EQUATIONS

Excerpts

EQUATIONS
An **equation** is an expression of the equality of two numbers. The parts of the expression separated by the sign of equality are called the **members** of the equation.
EQUATIONS
In each of these illustrations $b$ has been transposed (changed over) from the first to the second member, and in each case its sign has been changed.
EQUATIONS
***Any term may be transposed from one member of an equation to the other provided its sign be changed.***
EQUATIONS
*To clear an equation of fractions, multiply each member of the equation by the L.C.M. of the denominators of the fractional terms.*
EQUATIONS
***Any changes may be made in an equation which do not destroy the equality of the members.***
EQUATIONS
*To solve an equation, clear of fractions if necessary, transpose the terms containing the unknown number to one member and the known terms to the other, unite the terms, and divide both members by the coefficient of the unknown number*.

Equations

EQUATIONS
x = a - b
Subtracting b from each member of x + b = a gives x equal to a minus b, so b has been transposed with its sign changed.
EQUATIONS
x = c + b
Adding b to each member of x - b = c gives x equal to c plus b, so b has been transposed with its sign changed.

Problems

Exercise 53

Exercise 53, problem 1
$22-6x=34-12x$.
Printed answer:
- $x = 2$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 2
On the STU-32 (STU, rpn):
```
34 ENTER 22 −
```
```
ENTER 12 ENTER 6 −
```
```
÷
```
Calculator: +2E+0; the book prints 2. Run on the calculator core at firmware 628c96c.
Exercise 53, problem 10
$21x-57=6x-14x+30$.
Printed answer:
- $x = 3$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 3
On the STU-32 (STU, rpn):
```
30 ENTER 57 +
```
```
21 ENTER 6 ENTER 14 − −
```
```
÷
```
Calculator: +3E+0; the book prints 3. Run on the calculator core at firmware 628c96c.
Exercise 53, problem 11
$5x-27-11x+16=98-40x-41$.
Printed answer:
- $x = 2$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 2
On the STU-32 (STU, rpn):
```
98 ENTER 41 − 27 ENTER 16 − +
```
```
5 ENTER 11 − 40 +
```
```
÷
```
Calculator: +2E+0; the book prints 2. Run on the calculator core at firmware 628c96c.
Exercise 53, problem 12
$14(x-2)+3(x+1)= 2(x-5)$.
Printed answer:
- $x = 1$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 1
On the STU-32 (STU, rpn):
```
5 ENTER 2 × +/− 14 ENTER 2 × + 3 ENTER 1 × −
```
```
ENTER 14 ENTER 3 + 2 − ÷
```
Calculator: +1E+0; the book prints 1. Run on the calculator core at firmware 628c96c.
Exercise 53, problem 13
$6(23-x)-3x=3(4x-27)$.
Printed answer:
- $x = 10\frac{3}{7}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 73/7
On the STU-32 (STU, rpn):
```
6 ENTER 23 × 3 ENTER 27 × +
```
```
ENTER 6 ENTER 3 + 3 ENTER 4 × +
```
```
÷
```
Calculator: +1042857142857142857142857142857143E-32; the book prints 73/7. Run on the calculator core at firmware 628c96c.
Exercise 53, problem 14
$3(x-1)-2(x-3)+(x-2)-5=0$.
Printed answer:
- $x = 2$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 2
On the STU-32 (STU, rpn):
```
3 ENTER 1 × +/−
```
```
2 ENTER 3 × +
```
```
2 −
```
```
5 −
```
```
+/−
```
```
3 ENTER 2 − 1 + ÷
```
Calculator: +2E+0; the book prints 2. Run on the calculator core at firmware 628c96c.
Exercise 53, problem 15
$(x+5)(x-3)=(x+2)(x-5)$.
Printed answer:
- $x = 1$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 1
On the STU-32 (STU, rpn):
```
5 ENTER 3 × 2 ENTER 5 × −
```
```
2 ENTER 3 + ÷
```
Calculator: +1E+0; the book prints 1. Run on the calculator core at firmware 628c96c.
Exercise 53, problem 16
$(x+4)(x+7)=(x+2)(x+11)$.
Printed answer:
- $x = 3$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 3
On the STU-32 (STU, rpn):
```
2 ENTER 11 ×
```
```
4 ENTER 7 × −
```
```
4 ENTER 7 +
```
```
2 ENTER 11 + −
```
```
÷
```
Calculator: +3E+0; the book prints 3. Run on the calculator core at firmware 628c96c.
Exercise 53, problem 17
$(x-1)(x+4)(x-2)=x(x-2)(x+2)$.
Printed answer:
- $x = 4$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 4
Exercise 53, problem 18
$(x-5)(x+3)-(x-7)(x-2)-2(x-1)=-12$.
Printed answer:
- $x = 3$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 3
Exercise 53, problem 19
$(x+2)^{2}-(x-1)^{2}=5(2x+3)$.
Printed answer:
- $x = -3$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes -3
Exercise 53, problem 2
$5x-4=10x+ 11$.
Printed answer:
- $x = -3$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes -3
On the STU-32 (STU, rpn):
```
11 ENTER 4 +
```
```
5 ENTER 10 − ÷
```
Calculator: -3E+0; the book prints -3. Run on the calculator core at firmware 628c96c.
Exercise 53, problem 20
$(2x+3)(x+3)-14=(2x+1)(x+1)$.
Printed answer:
- $x = 1$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 1
On the STU-32 (STU, rpn):
```
3 ENTER 3 × 14 −
```
```
1 ENTER 1 × x↔y −
```
```
2 ENTER 3 × 3 +
```
```
2 ENTER 1 × 1 + −
```
```
÷
```
Calculator: +1E+0; the book prints 1. Run on the calculator core at firmware 628c96c.
Exercise 53, problem 21
$(x+1)^{2}+(x-5)^{2}=2(x+5)^{2}$.
Printed answer:
- $x = \frac{6}{7}$.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the printed answer does not match the problem 6/7
Exercise 53, problem 22
$7x - 15 + 4x - 6 = 4x - 9 - 9x$.
Printed answer:
- $x = \frac{3}{4}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 3/4
On the STU-32 (STU, rpn):
```
15 ENTER 6 + 9 −
```
```
7 ENTER 4 + 9 ENTER 4 − +
```
```
÷
```
Calculator: +75E-2; the book prints 3/4. Run on the calculator core at firmware 628c96c.
Exercise 53, problem 23
Divide the number 105 into three parts, such that the second shall be 5 more than the first, and the third three times the second.
Printed answer:
- $17$; $22$; $66$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {x: 17, y: 22, z: 66}
Exercise 53, problem 24
A man had a certain amount of money; he earned four times as much the next week, and found $30. If he then had seven times as much as at first, how much had he at first?
Printed answer:
- $\$ 15$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation 15
On the STU-32 (STU, rpn):
```
30 ENTER 7 ENTER 1 ENTER 4 + − ÷
```
Calculator: +15E+0; the book prints 15. Run on the calculator core at firmware 628c96c.
Exercise 53, problem 25
How many fourths are there in $7x$?
Printed answer:
- $28x$ fourths.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 28*x
Exercise 53, problem 26
How long will it take a man to build $x$ yards of wall if he builds $z$ feet a day?
Printed answer:
- $\displaystyle \frac{3x}{z}$ days.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 3*x/z
Exercise 53, problem 27
$\displaystyle 6x - \frac{x + 4}{3} = \frac{x}{3} + 28$.
Printed answer:
- $x = 5\frac{1}{2}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 11/2
On the STU-32 (STU, rpn):
```
3 ENTER 28 ×
```
```
4 +
```
```
6 ENTER 3 × 1 − 1 −
```
```
÷
```
Calculator: +55E-1; the book prints 11/2. Run on the calculator core at firmware 628c96c.
Exercise 53, problem 28
$\displaystyle \frac{2x - 1}{5} - \frac{x + 12}{3} - 4\frac{4}{5} = \frac{2x}{3}$.
Printed answer:
- $x = -15$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes -15
On the STU-32 (STU, rpn):
```
1 ENTER 5 ÷ +/−
```
```
12 ENTER 3 ÷ −
```
```
4 ENTER 5 × 4 + 5 ÷ −
```
```
2 ENTER 5 ÷ 1 ENTER 3 ÷ − 2 ENTER 3 ÷ −
```
```
÷ +/−
```
Calculator: -15E+0; the book prints -15. Run on the calculator core at firmware 628c96c.
Exercise 53, problem 29
$\displaystyle x - \frac{x}{4} + 25 = \frac{x}{3} + \frac{x}{2} + 21$.
Printed answer:
- $x = 48$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 48
On the STU-32 (STU, rpn):
```
21 ENTER 25 −
```
```
1 ENTER 4 1/x −
```
```
3 1/x −
```
```
2 1/x −
```
```
÷
```
Calculator: +4800000000000000000000000000000002E-32; the book prints 48. Run on the calculator core at firmware 628c96c.
Exercise 53, problem 3
$23-8x=80-11x$.
Printed answer:
- $x = 19$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 19
On the STU-32 (STU, rpn):
```
11 ENTER 8 −
```
```
80 ENTER 23 −
```
```
x↔y
```
```
÷
```
Calculator: +19E+0; the book prints 19. Run on the calculator core at firmware 628c96c.
Exercise 53, problem 30
$\displaystyle \frac{x - 3}{3} + \frac{5}{21} = -\frac{x - 8}{7}$.
Printed answer:
- $x = 4$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 4
On the STU-32 (STU, rpn):
```
8 ENTER 3 ×
```
```
21 ENTER 5 − +
```
```
7 ENTER 3 + ÷
```
Calculator: +4E+0; the book prints 4. Run on the calculator core at firmware 628c96c.
Exercise 53, problem 31
$\displaystyle \frac{8 - 5x}{12} + \frac{5x - 6}{4} - \frac{7x + 5}{6} = 0$.
Printed answer:
- $x = -5$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes -5
On the STU-32 (STU, rpn):
```
8 ENTER 12 ÷ 6 ENTER 4 ÷ − 5 ENTER 6 ÷ −
```
```
5 ENTER 4 ÷ 5 ENTER 12 ÷ +/− + 7 ENTER 6 ÷ −
```
```
÷ +/−
```
Calculator: -4999999999999999999999999999999996E-33; the book prints -5. Run on the calculator core at firmware 628c96c.
Exercise 53, problem 32
$\displaystyle \frac{3 + 5x}{4} + \frac{x + 2}{2} = 1\frac{5}{9}$.
Printed answer:
- $x = -\frac{1}{9}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes -1/9
Exercise 53, problem 33
$\displaystyle \frac{2x - 1}{3} - \frac{13}{42} = \frac{5x - 4}{6}$.
Printed answer:
- $x = \frac{1}{7}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 1/7
On the STU-32 (STU, rpn):
```
1 ENTER 3 ÷ 13 ENTER 42 ÷ −
```
```
2 ENTER 3 ÷ 5 ENTER 6 ÷ −
```
```
÷ +/−
```
Calculator: +1428571428571428571428571428571429E-34; the book prints 1/7. Run on the calculator core at firmware 628c96c.
Exercise 53, problem 34
$\displaystyle \frac{1 - 11x}{7} - \frac{7x}{13} = \frac{2}{13} - \frac{8x - 15}{3}$.
Printed answer:
- $x = 9$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 9
On the STU-32 (STU, rpn):
```
2 ENTER 13 ÷ 15 ENTER 3 ÷ + 1 ENTER 7 ÷ −
```
```
8 ENTER 3 ÷ 11 ENTER 7 ÷ − 7 ENTER 13 ÷ −
```
```
÷
```
Calculator: +8999999999999999999999999999999988E-33; the book prints 9. Run on the calculator core at firmware 628c96c.
Exercise 53, problem 35
$\displaystyle \frac{2}{x}-\frac{3}{2x} = \frac{7}{24}-\frac{2}{3x}$.
Printed answer:
- $x = 4$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 4
On the STU-32 (STU, rpn):
```
24 ENTER 2 ×
```
```
24 ENTER 3 × 2 ÷ −
```
```
24 ENTER 2 × 3 ÷ +
```
```
7 ÷
```
Calculator: +4E+0; the book prints 4. Run on the calculator core at firmware 628c96c.
Exercise 53, problem 36
$\displaystyle \frac{1}{4}+\frac{1}{3x} = \frac{11}{36}+\frac{1}{6x}$.
Printed answer:
- $x = 3$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 3
On the STU-32 (STU, rpn):
```
11 ENTER 36 ÷ 1 ENTER 4 ÷ −
```
```
1/x
```
```
6 ÷
```
Calculator: +2999999999999999999999999999999998E-33; the book prints 3. Run on the calculator core at firmware 628c96c.
Exercise 53, problem 37
$\displaystyle \frac{3}{4x}-\frac{2}{x}+\frac{x-2}{2x}+7\frac{5}{12} = 5+\frac{4}{6x}$.
Printed answer:
- $x = 1$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 1
On the STU-32 (STU, rpn):
```
4 ENTER 12 × 6 ÷
```
```
2 ENTER 12 × +
```
```
12 +
```
```
3 ENTER 12 × 4 ÷ −
```
```
7 ENTER 12 × 5 +
```
```
12 ENTER 2 ÷ +
```
```
12 ENTER 5 × −
```
```
÷
```
Calculator: +1E+0; the book prints 1. Run on the calculator core at firmware 628c96c.
Exercise 53, problem 38
$\displaystyle \frac{2x+3}{5x}-\frac{3}{x}+4 = \frac{1}{2x}+\frac{3}{4x}+2\frac{23}{40}$.
Printed answer:
- $x = 2$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 2
On the STU-32 (STU, rpn):
```
1 ENTER 2 ÷ 3 ENTER 4 ÷ + 3 + 3 ENTER 5 ÷ −
```
```
ENTER 2 ENTER 5 ÷ 4 + 103 ENTER 40 ÷ − ÷
```
Calculator: +2E+0; the book prints 2. Run on the calculator core at firmware 628c96c.
Exercise 53, problem 39
$\displaystyle \frac{x+9}{11}-\frac{2-x}{5} = \frac{x+5}{7}$.
Printed answer:
- $x = 2$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 2
On the STU-32 (STU, rpn):
```
1 ENTER 11 ÷ 1 ENTER 5 ÷ + 1 ENTER 7 ÷ −
```
```
1/x
```
```
5 ENTER 7 ÷ 2 ENTER 5 ÷ + 9 ENTER 11 ÷ −
```
```
×
```
Calculator: +1999999999999999999999999999999999E-33; the book prints 2. Run on the calculator core at firmware 628c96c.
Exercise 53, problem 4
$5x-21=7x + 5$.
Printed answer:
- $x = -13$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes -13
On the STU-32 (STU, rpn):
```
5 ENTER 7 −
```
```
5 ENTER 21 +/− −
```
```
x↔y
```
```
÷
```
Calculator: -13E+0; the book prints -13. Run on the calculator core at firmware 628c96c.
Exercise 53, problem 40
$\displaystyle \frac{x+4}{5}-\frac{4-x}{7} = \frac{x+1}{3}$.
Printed answer:
- $x = 11$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 11
On the STU-32 (STU, rpn):
```
7 ENTER 3 × 4 ×
```
```
5 ENTER 3 × 4 × −
```
```
5 ENTER 7 × x↔y −
```
Calculator: +11E+0; the book prints 11. Run on the calculator core at firmware 628c96c.
Exercise 53, problem 41
$\displaystyle \frac{2}{5}(x-6)-\frac{3}{16}(x-1) = \frac{5}{12}(4-x)-\frac{5}{48}$.
Printed answer:
- $x = 6$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 6
On the STU-32 (STU, rpn):
```
5 ENTER 12 ÷ 4 × 5 ENTER 48 ÷ − 2 ENTER 5 ÷ 6 × + 3 ENTER 16 ÷ −
```
```
2 ENTER 5 ÷ 3 ENTER 16 ÷ − 5 ENTER 12 ÷ + ÷
```
Calculator: +6000000000000000000000000000000000E-33; the book prints 6. Run on the calculator core at firmware 628c96c.
Exercise 53, problem 42
$\displaystyle \frac{4+x}{4}-\frac{1-x}{7}-\frac{1}{5}(8-x) = \frac{x-23}{5}+7$.
Printed answer:
- $x = 8$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 8
Exercise 53, problem 43
How long will it take a man to walk $x$ miles if he walks 15 miles in $b$ hours?
Printed answer:
- $\displaystyle \frac{bx}{15}$ hrs.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles b*x/15
Exercise 53, problem 44
What is the interest on $m$ dollars for one year at 5 per cent?
Printed answer:
- $\displaystyle \frac{5m}{100}$ dols.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 5*m/100
Exercise 53, problem 45
What are the two numbers whose sum is 57, and whose difference is 25?
Printed answer:
- $16$; $41$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {x: 41, y: 16}
Exercise 53, problem 46
What is the interest on $b$ dollars for $y$ years at 4 per cent?
Printed answer:
- $\displaystyle \frac{4by}{100}$ dols.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 4*b*y/100
Exercise 53, problem 47
$(c+a)x+(c-b)x = c^2$.
Printed answer:
- $\displaystyle x = \frac{c^2}{ a - b + 2c }$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes c**2/(a - b + 2*c)
Exercise 53, problem 48
$(a-b)x+(a+b)x = a$.
Printed answer:
- $x = \frac{1}{2}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 1/2
Exercise 53, problem 49
$2x+a(x-2) = a+6$.
Printed answer:
- $x = 3$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 3
Exercise 53, problem 5
$18x-43=17-6x$.
Printed answer:
- $x = 2\frac{1}{2}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 5/2
On the STU-32 (STU, rpn):
```
17 ENTER 43 +
```
```
18 ENTER 6 +
```
```
÷
```
Calculator: +25E-1; the book prints 5/2. Run on the calculator core at firmware 628c96c.
Exercise 53, problem 50
$b(2x-a)-a^2 = 2x(a+b)-3ab$.
Printed answer:
- $\displaystyle x = \frac{2b - a}{2}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes (2*b - a)/2
Exercise 53, problem 51
$\displaystyle a^2+c^2=\frac{cx}{a}+\frac{ax}{c}$.
Printed answer:
- $\displaystyle x = \frac{ac(a^2 + c^2)}{a + c}$.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the printed answer does not match the problem a*c*(a**2 + c**2)/(a + c)
Exercise 53, problem 52
$b^4-x^2+2bx=(b^2+x)(b^2-x)$.
Printed answer:
- $x = 0$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 0
Exercise 53, problem 53
$x^2+4a^2+a^4=\left(x+a^2\right)^2$.
Printed answer:
- $x = 2$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 2
Exercise 53, problem 54
$b^2(x-b)+a^2(x-a)=abx$.
Printed answer:
- $x = a + b$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes a + b
Exercise 53, problem 55
$\displaystyle \frac{2x+5}{5x+3}=\frac{2x-4}{5x-6}$.
Printed answer:
- $x = \frac{2}{3}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 2/3
Exercise 53, problem 56
$\displaystyle \frac{6x+5}{2x-3}=\frac{3x-4}{x+1}$.
Printed answer:
- $x = \frac{1}{4}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 1/4
On the STU-32 (STU, rpn):
```
4 +/− ENTER 3 +/− ×
```
```
5 ENTER 1 × −
```
```
6 ENTER 1 × 5 ENTER 1 × +
```
```
3 ENTER 3 +/− × −
```
```
2 ENTER 4 +/− × −
```
```
÷
```
Calculator: +25E-2; the book prints 1/4. Run on the calculator core at firmware 628c96c.
Exercise 53, problem 57
$\displaystyle \frac{2+9x}{3(6x+7)}=\frac{2x+9}{35+4x}$.
Printed answer:
- $x = 1$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 1
Exercise 53, problem 58
$\displaystyle \frac{3}{2-5x}-\frac{4}{1-3x}=0$.
Printed answer:
- $x = \frac{5}{11}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 5/11
On the STU-32 (STU, rpn):
```
4 ENTER 2 × 3 ENTER 1 × −
```
```
4 ENTER 5 × 3 ENTER 3 × −
```
```
÷
```
Calculator: +4545454545454545454545454545454545E-34; the book prints 5/11. Run on the calculator core at firmware 628c96c.
Exercise 53, problem 59
$\displaystyle \frac{2(3x+4)}{1+2x}-1=\frac{2(19+x)}{x+12}$.
Printed answer:
- $x = 2$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 2
Exercise 53, problem 6
$18-8x=12x- 87$.
Printed answer:
- $x = 5\frac{1}{4}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 21/4
On the STU-32 (STU, rpn):
```
18 ENTER 87 +
```
```
8 ENTER 12 +
```
```
÷
```
Calculator: +525E-2; the book prints 21/4. Run on the calculator core at firmware 628c96c.
Exercise 53, problem 60
$\displaystyle 1-\frac{x}{x+2}=\frac{4}{x+6}$.
Printed answer:
- $x = 2$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 2
On the STU-32 (STU, rpn):
```
4 ENTER 2 × 2 ENTER 6 × −
```
```
2 ENTER 4 − ÷
```
Calculator: +2E+0; the book prints 2. Run on the calculator core at firmware 628c96c.
Exercise 53, problem 61
$\displaystyle \frac{x^2}{1-x^2}+\frac{x+1}{x-1}=\frac{3}{x+1}$.
Printed answer:
- $x = 4$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 4
Exercise 53, problem 62
$\displaystyle \frac{2+3x}{1-x}+5=\frac{2x-4}{x+2}$.
Printed answer:
- $x = 6$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 6
Exercise 53, problem 63
$\displaystyle \frac{5}{6+2x}+\frac{2}{x+1}=\frac{2\frac{1}{2}}{2+2x}+\frac{4}{3+x}$.
Printed answer:
- $x = 1$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 1
Exercise 53, problem 64
$\displaystyle \frac{2(1-2x)}{1-3x}+\frac{1}{6}=\frac{1-3x}{1-2x}$.
Printed answer:
- $x = \frac{7}{17}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 7/17
Exercise 53, problem 65
$\displaystyle \frac{x-8}{x-6}-\frac{x}{x-2}=\frac{x-9}{x-7}-\frac{x+1}{x-1}$.
Printed answer:
- $x = 4$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 4
Exercise 53, problem 66
$\displaystyle \frac{x+5}{x+8} - \frac{x+6}{x+9}=\frac{x+2}{x+5}-\frac{x+3}{x+6}$.
Printed answer:
- $x = -7$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes -7
Exercise 53, problem 67
Find three consecutive numbers whose sum is 81.
Printed answer:
- $26$; $27$; $28$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {x: 26}
On the STU-32 (STU, rpn):
```
81 ENTER 1 ENTER 2 + −
```
```
3 ÷
```
Calculator: +26E+0; the book prints 26. Run on the calculator core at firmware 628c96c.
Exercise 53, problem 68
A’s age is double B’s, B’s is three times C’s, and C is $y$ years old. What is A’s age?
Printed answer:
- $6y$ years.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 6*y
Exercise 53, problem 69
How many men will be required to do in $a$ hours what $x$ men do in 6 hours?
Printed answer:
- $\displaystyle \frac{6x}{a}$ men.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation 6*x/a
Exercise 53, problem 7
$9x-(2x-5)=4x+(13 + x)$.
Printed answer:
- $x = 4$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 4
On the STU-32 (STU, rpn):
```
13 ENTER 5 −
```
```
9 ENTER 2 − 4 − 1 −
```
```
÷
```
Calculator: +4E+0; the book prints 4. Run on the calculator core at firmware 628c96c.
Exercise 53, problem 70
Find the sum of three consecutive odd numbers of which the middle one is $4x + 1$.
Printed answer:
- $12x + 3$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 12*x + 3
Exercise 53, problem 8
$15x-2(5x-4)-39 = 0$.
Printed answer:
- $x = 6\frac{1}{5}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 31/5
On the STU-32 (STU, rpn):
```
39 ENTER 2 ENTER 4 × −
```
```
15 ENTER 2 ENTER 5 × − ÷
```
Calculator: +62E-1; the book prints 31/5. Run on the calculator core at firmware 628c96c.
Exercise 53, problem 9
$12x-18x+17=8x+3$.
Printed answer:
- $x = 1$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 1
On the STU-32 (STU, rpn):
```
3 ENTER 17 −
```
```
12 ENTER 18 − 8 −
```
```
÷
```
Calculator: +1E+0; the book prints 1. Run on the calculator core at firmware 628c96c.

Exercise 54

Exercise 54, problem 1
In a school of 836 pupils there is one boy to every three girls. How many are there of each?
Printed answer:
- $209$ boys; $627$ girls.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"b": "209", "g": "627"}
Exercise 54, problem 10
If I add 18 to a certain number, five times this second number will equal eleven times the original number. What is the original number?
Printed answer:
- $15$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"x": "15"}
Exercise 54, problem 11
In a mixture of 48 pounds of coffee there is one-third as much Mocha as Java. How much is there of each?
Printed answer:
- $12$ M.; $36$ J.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"J": "36", "M": "12"}
Exercise 54, problem 12
The half and fifth of a number are together equal to 56. What is the number?
Printed answer:
- $80$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"x": "80"}
Exercise 54, problem 13
What number increased by one-third and one-fourth of itself, and 7 more, equals 45?
Printed answer:
- $24$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"x": "24"}
Exercise 54, problem 14
What number is doubled by adding to it three-eighths of itself, one-third of itself, and 14?
Printed answer:
- $48$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"x": "48"}
Exercise 54, problem 15
A grocer sold 27 pounds of sugar, tea, and meal. Of meal he sold 3 pounds more than of tea, and of sugar 6 pounds more than of meal. How many pounds of each did he sell?
Printed answer:
- t., $5$ lbs.; m., $8$ lbs.; s., $14$ lbs.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the record may be misread {"m": "8", "s": "14", "t": "5"}
Exercise 54, problem 16
A son is two-sevenths as old as his father. If the sum of their ages is 45 years, how old is each?
Printed answer:
- $35$; $10$ yrs.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the record may be misread {"f": "35", "s": "10"}
Exercise 54, problem 17
Two men invest $2990 in business, one putting in four-ninths as much as the other. How much does each invest?
Printed answer:
- $2070, $920.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"a": "2070", "b": "920"}
Exercise 54, problem 18
In an election 47,519 votes were cast for three candidates. One candidate received 2061 votes less, and the other 1546 votes less, than the winning candidate. How many votes did each receive?
Printed answer:
- $17,042$; $14,981$; $15,496$.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the record may be misread {"s": "14981", "t": "15496", "w": "17042"}
Exercise 54, problem 19
John had twice as many stamps as Ralph, but after he had bought 65, and Ralph had lost 16, they found that they had together 688. How many had each at first?
Printed answer:
- R., $213$; J., $426$.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the record may be misread {"J": "426", "R": "213"}
Exercise 54, problem 2
Divide 253 into three parts, so that the first part shall be four times the second, and the second twice the third.
Printed answer:
- $184$; $46$; $23$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"x": "184", "y": "46", "z": "23"}
Exercise 54, problem 20
Find three consecutive numbers whose sum is 192.
Printed answer:
- $63$; $64$; $65$.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the record may be misread {"a": "63", "b": "64", "c": "65"}
Exercise 54, problem 21
If 17 be added to the sum of two numbers whose difference is 12, the result will be 61. What are the numbers?
Printed answer:
- $16$; $28$.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the record may be misread {"x": "28", "y": "16"}
Exercise 54, problem 22
Divide 120 into two parts such that five times one part may be equal to three times the other.
Printed answer:
- $45$; $75$.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the record may be misread {"p": "45", "q": "75"}
Exercise 54, problem 23
Mr. Johnson is twice as old as his son; 12 years ago he was three times as old. What is the age of each?
Printed answer:
- $24$; $48$ yrs.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the record may be misread {"J": "48", "s": "24"}
Exercise 54, problem 24
Henry is six times as old as his sister, but in 3 years from now he will be only three times as old. How old is each?
Printed answer:
- $2$; $12$ yrs.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the record may be misread {"H": "12", "S": "2"}
Exercise 54, problem 25
Samuel is 16 years older than James; 4 years ago he was three times as old. How old is each?
Printed answer:
- J., $12$ yrs.; S., $28$ yrs.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the record may be misread {"J": "12", "S": "28"}
Exercise 54, problem 26
Martha is 5 years old and her father is 30. In how many years will her father be twice as old as Martha?
Printed answer:
- $20$ yrs.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the record may be misread {"x": "20"}
Exercise 54, problem 27
George is three times as old as Amelia; in 6 years his age will be twice hers. What is the age of each?
Printed answer:
- A., $6$ yrs.; G., $18$ yrs.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the record may be misread {"A": "6", "G": "18"}
Exercise 54, problem 28
Esther is three-fourths as old as Edward; 20 years ago she was half as old. What is the age of each?
Printed answer:
- Ed., $40$ yrs.; Es., $30$ yrs.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the record may be misread {"D": "40", "E": "30"}
Exercise 54, problem 29
Mary is 4 years old and Flora is 9. In how many will Mary be two-thirds as old as Flora?
Printed answer:
- $6$ yrs.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the record may be misread {"x": "6"}
Exercise 54, problem 3
The sum of the ages of two brothers is 44 years, and one of them is 12 years older than the other. Find their ages.
Printed answer:
- $16$; $28$ yrs
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"o": "28", "y": "16"}
Exercise 54, problem 30
Harry is 9 years older than his little brother; in 6 years he will be twice as old. How old is each?
Printed answer:
- $3$; $12$ yrs.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the record may be misread {"H": "12", "b": "3"}
Exercise 54, problem 31
Divide $2x^4+27xy^3-81y^4$ by $x+3y$.
Printed answer:
- $2 x^3 -6 x^2y + 18xy^2 - 27 y^3$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*x**3 - 6*x**2*y + 18*x*y**2 - 27*y**3
Exercise 54, problem 32
Prove $\left(a^2+ab+b^2\right)^2-\left(a^2-ab+b^2\right)^2=4ab(a^2+b^2)$.
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise 54, problem 33
Find the value of $\displaystyle \frac{x-y}{y}+\frac{2x}{x-y}-\frac{x^3+x^2y}{x^2y-y^3}$.
Printed answer:
- $\displaystyle \frac{y}{x - y}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes y/(x - y)
Exercise 54, problem 34
Solve $\displaystyle \frac{5x-7}{2}-3x=\frac{2x+7}{3}-14$.
Printed answer:
- $x = 7$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 7
On the STU-32 (STU, rpn):
```
7 ENTER 3 ÷ 14 − 7 ENTER 2 ÷ +
```
```
5 ENTER 2 ÷ 3 − 2 ENTER 3 ÷ − ÷
```
Calculator: +7000000000000000000000000000000001E-33; the book prints 7. Run on the calculator core at firmware 628c96c.
Exercise 54, problem 35
Mr. Ames has $132, and Mr. Jones $43. How much must Mr. A. give to Mr. J. so that Mr. J. may have three-fourths as much as Mr. A.?
Printed answer:
- $32.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"x": "32"}
Exercise 54, problem 36
A has $101, and B has $35; each loses a certain sum, and then A has four times as much as B. What was the sum lost by each?
Printed answer:
- $13.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"x": "13"}
Exercise 54, problem 37
A certain sum of money was divided among A, B, and C; A and B received $75, A and C $108, and B and C $89. How much did each receive? *Suggestion*. Let $x$ equal what A received.
Printed answer:
- A, $47; B, $28; C, $61.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"A": "47", "B": "28", "C": "61"}
Exercise 54, problem 38
Mary and Jane have the same amount of money. If Mary should give Jane 40 cents, she would have one-third as much as Jane. What amount of money has each?
Printed answer:
- 80 cts.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"m": "80"}
Exercise 54, problem 39
An ulster and a suit of clothes cost $43; the ulster and a hat cost $27; the suit of clothes and the hat cost $34. How much did each cost?
Printed answer:
- u., $18; cl., $25; h., $9.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"h": "9", "s": "25", "u": "18"}
Exercise 54, problem 4
Find two numbers whose sum is 158, and whose difference is 86.
Printed answer:
- $36$; $122$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"x": "122", "y": "36"}
Exercise 54, problem 40
John, Henry, and Arthur picked berries, and sold them; John and Henry received $4.22, John and Arthur $3.05, Henry and Arthur $3.67. How much did each receive for his berries?
Printed answer:
- J., $ 1.80; H., $2.42; A., $1.25
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"A": "5/4", "H": "121/50", "J": "9/5"}
Exercise 54, problem 41
A can do a piece of work in 4 days, and B can do it in 6 days. In what time can they do it working together? *Suggestion*. Let $x$ equal the required time. Then find what part of the work each can do in one day.
Printed answer:
- $2\frac{2}{5}$ days.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"x": "12/5"}
Exercise 54, problem 42
Mr. Brown can build a stone wall in 10 days, and Mr. Mansfield in 12 days. How long would it take them to do it working together?
Printed answer:
- $5\frac{5}{11}$ days.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"x": "60/11"}
Exercise 54, problem 43
Mr. Richards and his son can hoe a field of corn in 9 hours, but it takes Mr. Richards alone 15 hours. How long would it take the son to hoe the field?
Printed answer:
- $22\frac{1}{2}$ hrs.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"s": "45/2"}
Exercise 54, problem 44
A can do a piece of work in 4 hours, B can do it in 6 hours, and C in 3 hours. How long would it take them working together?
Printed answer:
- $1\frac{1}{3}$ hrs.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"x": "4/3"}
Exercise 54, problem 45
A can mow a field in 6 hours, B in 8 hours, and with the help of C they can do it in 2 hours. How long would it take C working alone?
Printed answer:
- $4\frac{4}{5}$ hrs.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"c": "24/5"}
Exercise 54, problem 46
A tank can be emptied by two pipes in 5 hours and 7 hours respectively. In what time can it be emptied by the two pipes together?
Printed answer:
- $2\frac{11}{12}$ hrs.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"x": "35/12"}
Exercise 54, problem 47
A cistern can be filled by two pipes in 4 hours and 6 hours respectively, and can be emptied by a third in 15 hours. In what time could the cistern be filled if all three pipes were running?
Printed answer:
- $2\frac{6}{7}$ hrs.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"x": "20/7"}
Exercise 54, problem 48
A and B together can do a piece of work in 8 days, A and C together in 10 days, and A by himself in 12 days. In what time can B and C do it? In what time can A, B, and C together do it?
Printed answer:
- $17\frac{1}{7}$; $7\frac{1}{17}$ days.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles ["120/7", "120/17"]
Exercise 54, problem 49
John and Henry can together paint a fence in 2 hours, John and Lewis together in 4 hours, and John by himself in 6 hours. In what time can the three together do the painting?
Printed answer:
- $1\frac{5}{7}$ hrs.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"x": "12/7"}
Exercise 54, problem 5
Henry and Susan picked 16 quarts of berries. Henry picked 4 quarts less than three times as many as Susan. How many quarts did each pick?
Printed answer:
- S., $5$; H., $11$ qts.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"H": "11", "S": "5"}
Exercise 54, problem 50
C can do a piece of work in $a$ days, and D can do the same work in $b$ days. In how many days can they do it working together?
Printed answer:
- $\displaystyle \frac{ab}{a + b}$ days.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation a*b/(a + b)
Exercise 54, problem 51
James and Thomas can do a piece of work in $d$ days and James alone can do it in $c$ days. How long would it take Thomas alone?
Printed answer:
- $\displaystyle \frac{cd}{c - d}$ days.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation c*d/(c - d)
Exercise 54, problem 52
Solve $\displaystyle \frac{3+x}{3-x}-\frac{1+x}{1-x}-\frac{2+x}{2-x}=1$.
Printed answer:
- $x = 1\frac{1}{2}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 3/2
Exercise 54, problem 53
Find the value of $\displaystyle \frac{x^2}{(x-y)(x-z)}+\frac{y^2}{(y-x)(y-z)}+\frac{z^2}{(z-x)(z-y)}$.
Printed answer:
- $1$.
unverified: no computed check settled this one (yet)
How it was checked
- identity: the record may be misread 1
Exercise 54, problem 54
Expand $(x^2-2)(x^2+2)(x^2+3)(x^2-3)$.
Printed answer:
- $x^8 - 13x^4 + 36$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**8 - 13*x**4 + 36
Exercise 54, problem 55a
Factor $9a^2+12ab+4b^2$, $12+7x+x^2$, $ac-2bc-3ad+6bd$.
Printed answer:
- $(3a + 2b)^2$; $(3 + x)(4 + x)$; $(a - 2b)(c - 3d)$.
verified: the printed answer passed a computed check
How it was checked
- factor: passes (3*a + 2*b)**2
Exercise 54, problem 55b
Factor $9a^2+12ab+4b^2$, $12+7x+x^2$, $ac-2bc-3ad+6bd$.
Printed answer:
- $(3a + 2b)^2$; $(3 + x)(4 + x)$; $(a - 2b)(c - 3d)$.
verified: the printed answer passed a computed check
How it was checked
- factor: passes (3 + x)*(4 + x)
Exercise 54, problem 55c
Factor $9a^2+12ab+4b^2$, $12+7x+x^2$, $ac-2bc-3ad+6bd$.
Printed answer:
- $(3a + 2b)^2$; $(3 + x)(4 + x)$; $(a - 2b)(c - 3d)$.
verified: the printed answer passed a computed check
How it was checked
- factor: passes (a - 2*b)*(c - 3*d)
Exercise 54, problem 56a
At what time are the hands of a clock together between 2 and 3? Between 5 and 6? Between 9 and 10?
Printed answer:
- tabularll(1)& $10\frac{10}{11}$ m. (2)& $27\frac{3}{11}$ m. (3) &$49\frac{1}{11}$ m. tabular
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation 120/11
On the STU-32 (STU, rpn):
```
30 ENTER 2 ×
```
```
2 ×
```
```
11 ÷
```
Calculator: +1090909090909090909090909090909091E-32; the book prints 120/11. Run on the calculator core at firmware 628c96c.
Exercise 54, problem 56b
At what time are the hands of a clock together between 2 and 3? Between 5 and 6? Between 9 and 10?
Printed answer:
- tabularll(1)& $10\frac{10}{11}$ m. (2)& $27\frac{3}{11}$ m. (3) &$49\frac{1}{11}$ m. tabular
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation 300/11
On the STU-32 (STU, rpn):
```
5 ENTER 30 ×
```
```
2 ×
```
```
11 ÷
```
Calculator: +2727272727272727272727272727272727E-32; the book prints 300/11. Run on the calculator core at firmware 628c96c.
Exercise 54, problem 56c
At what time are the hands of a clock together between 2 and 3? Between 5 and 6? Between 9 and 10?
Printed answer:
- tabularll(1)& $10\frac{10}{11}$ m. (2)& $27\frac{3}{11}$ m. (3) &$49\frac{1}{11}$ m. tabular
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation 540/11
On the STU-32 (STU, rpn):
```
9 ENTER 30 ×
```
```
2 ×
```
```
11 ÷
```
Calculator: +4909090909090909090909090909090909E-32; the book prints 540/11. Run on the calculator core at firmware 628c96c.
Exercise 54, problem 57a
At what time are the hands of a clock at right angles between 2 and 3? Between 4 and 5? Between 7 and 8?
Printed answer:
- tabularrrcl(1)& $27\frac{3}{11}$ m. (2)& $5\frac{5}{11}$ m.; &$38\frac{2}{11}$ m. (3)& $21\frac{9}{11}$ m.; &$54\frac{6}{11}$ m. tabular
unverified: no computed check settled this one (yet)
How it was checked
- solve: the record may be misread ["150/11"]
Exercise 54, problem 57b
At what time are the hands of a clock at right angles between 2 and 3? Between 4 and 5? Between 7 and 8?
Printed answer:
- tabularrrcl(1)& $27\frac{3}{11}$ m. (2)& $5\frac{5}{11}$ m.; &$38\frac{2}{11}$ m. (3)& $21\frac{9}{11}$ m.; &$54\frac{6}{11}$ m. tabular
unverified: no computed check settled this one (yet)
How it was checked
- solve: the record may be misread ["60/11", "420/11"]
Exercise 54, problem 57c
At what time are the hands of a clock at right angles between 2 and 3? Between 4 and 5? Between 7 and 8?
Printed answer:
- tabularrrcl(1)& $27\frac{3}{11}$ m. (2)& $5\frac{5}{11}$ m.; &$38\frac{2}{11}$ m. (3)& $21\frac{9}{11}$ m.; &$54\frac{6}{11}$ m. tabular
unverified: no computed check settled this one (yet)
How it was checked
- solve: the record may be misread ["120/11", "600/11"]
Exercise 54, problem 58a
At what time are the hands of a clock opposite each other between 3 and 4? Between 8 and 9? Between 12 and 1?
Printed answer:
- tabularll(1)& $49\frac{1}{11}$ m. (2) & $10\frac{10}{11}$ m. (3) & $32\frac{8}{11}$ m. tabular
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation 540/11
On the STU-32 (STU, rpn):
```
30 ENTER 3 × 180 + 2 × 11 ÷
```
Calculator: +4909090909090909090909090909090909E-32; the book prints 540/11. Run on the calculator core at firmware 628c96c.
Exercise 54, problem 58b
At what time are the hands of a clock opposite each other between 3 and 4? Between 8 and 9? Between 12 and 1?
Printed answer:
- tabularll(1)& $49\frac{1}{11}$ m. (2) & $10\frac{10}{11}$ m. (3) & $32\frac{8}{11}$ m. tabular
unverified: no computed check settled this one (yet)
How it was checked
- solve: the printed answer does not match the problem ["120/11", "780/11"]
Exercise 54, problem 58c
At what time are the hands of a clock opposite each other between 3 and 4? Between 8 and 9? Between 12 and 1?
Printed answer:
- tabularll(1)& $49\frac{1}{11}$ m. (2) & $10\frac{10}{11}$ m. (3) & $32\frac{8}{11}$ m. tabular
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation 360/11
Exercise 54, problem 59
At what times between 4 and 5 o’clock are the hands of a watch ten minutes apart?
Printed answer:
- $10\frac{10}{11}$ m., $32\frac{8}{11}$ m.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation ["120/11", "360/11"]
Exercise 54, problem 6
Divide 127 into three parts, such that the second shall be 5 more than the first, and the third four times the second.
Printed answer:
- $17$; $22$; $88$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"a": "17", "b": "22", "c": "88"}
Exercise 54, problem 60
At what time between 8 and 9 o’clock are the hands of a watch 25 minutes apart?
Printed answer:
- $16\frac{4}{11}$ m.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation 180/11
On the STU-32 (STU, rpn):
```
30 ENTER 8 × 150 − 2 × 11 ÷
```
Calculator: +1636363636363636363636363636363636E-32; the book prints 180/11. Run on the calculator core at firmware 628c96c.
Exercise 54, problem 61
At what time between 5 and 6 o’clock is the minute hand three minutes ahead of the hour hand?
Printed answer:
- $30\frac{6}{11}$ m.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation 336/11
On the STU-32 (STU, rpn):
```
18 ENTER 30 ENTER 5 × +
```
```
2 × 11 ÷
```
Calculator: +3054545454545454545454545454545455E-32; the book prints 336/11. Run on the calculator core at firmware 628c96c.
Exercise 54, problem 62
It was between 12 and 1 o’clock; but a man, mistaking the hour hand for the minute hand, thought that it was 55 minutes later than it really was. What time was it?
Printed answer:
- $5\frac{5}{11}$ m. past 12M.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the printed answer does not match the problem 60/11
Exercise 54, problem 63
At what time between 11 and 12 o’clock are the hands two minutes apart?
Printed answer:
- $57\frac{9}{11}$ m.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation 636/11
On the STU-32 (STU, rpn):
```
30 ENTER 11 × 12 −
```
```
2 × 11 ÷
```
Calculator: +5781818181818181818181818181818182E-32; the book prints 636/11. Run on the calculator core at firmware 628c96c.
Exercise 54, problem 64
A messenger who travels at the rate of 10 miles an hour is followed, 4 hours later, by another who travels at the rate of 12 miles an hour. How long will it take the second to overtake the first?
Printed answer:
- 20 hrs.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"t": "20"}
Exercise 54, problem 65
A courier who travels at the rate of 19 miles in 4 hours is followed, 8 hours later, by another who travels at the rate of 19 miles in 3 hours. In what time will the second overtake the first? How far will the first have gone before he is overtaken?
Printed answer:
- 24 hrs.; 152 hrs
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"d": "152", "t": "24"}
Exercise 54, problem 66
A train going at the rate of 20 miles an hour is followed, on a parallel track, 4 hours later, by an express train. The express overtakes the first train in $5\frac{1}{3}$ hours. What is the rate of the express train?
Printed answer:
- 35 miles.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"r": "35"}
Exercise 54, problem 67
A messenger started for Washington at the rate of $6\frac{1}{2}$ miles an hour. Six hours later a second messenger followed and in $4\frac{7}{8}$ hours overtook the first just as he was entering the city. At what rate did the second messenger go? How far was it to Washington?
Printed answer:
- $14\frac{1}{2}$ miles; $70\frac{11}{16}$ miles.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"d": "1131/16", "r": "29/2"}
Exercise 54, problem 68
How far could a man ride at the rate of 8 miles an hour so as to walk back at the rate of 4 miles an hour and be gone only 9 hours?
Printed answer:
- 24 miles.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"d": "24"}
Exercise 54, problem 69
Two persons start at 10 A.M. from towns A and B, $55\frac{1}{2}$ miles apart. The one starting from A walks at the rate of $4\frac{1}{4}$ miles an hour, but stops 2 hours on the way; the other walks at the rate of $3\frac{3}{4}$ miles an hour without stopping. When will they meet? How far will each have traveled? *Suggestion*. Let x equal the number of hours.
Printed answer:
- 6 P.M.; 30 miles from B; $25\frac{1}{2}$ miles from A.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"t": "8", "dA": "51/2", "dB": "30"}
Exercise 54, problem 7
Twice a certain number added to four times the double of that number is 90. What is the number?
Printed answer:
- $9$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"x": "9"}
Exercise 54, problem 70
A boy who runs at the rate of $12\frac{1}{2}$ yards per second, starts 16 yards behind another whose rate is 11 yards per second. How soon will the first boy be 8 yards ahead of the second?
Printed answer:
- 16 sec.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"t": "16"}
Exercise 54, problem 71
A rectangle whose length is 4 ft. more than its width would have its area increased 56 sq. ft. if its length and width were each made 2 ft. more. What are its dimensions?
Printed answer:
- 11 ft. by 15 ft.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"L": "15", "W": "11"}
Exercise 54, problem 72
The length of a room is double its width. If the length were 3 ft. less and the width 3 ft. more, the area would be increased 27 sq. ft. Find the dimensions of the room.
Printed answer:
- 12 ft. by 24 ft.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"L": "24", "W": "12"}
Exercise 54, problem 73
A floor is two-thirds as wide as it is long. If the width were 2 ft. more and the length 4 ft. less, the area would be diminished 22 sq. ft. What are its dimensions?
Printed answer:
- 14 ft. by 21 ft.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"L": "21", "W": "14"}
Exercise 54, problem 74
A rectangle has its length and width respectively 4 ft. longer and 2 ft. shorter than the side of an equivalent square. Find its area.
Printed answer:
- 16 sq. ft.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"area": "16"}
Exercise 54, problem 75
An enclosed garden is 24 ft. greater in length than in width. 684 sq. ft. is used for a walk 3 ft. wide extending around the garden inside the fence. How long is the garden?
Printed answer:
- 48 ft. by 72 ft.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"L": "72", "W": "48"}
Exercise 54, problem 76a
Factor $\displaystyle \frac{x^2}{4}-\frac{y^2z^4}{9m^6}$, $x^6-27y^3$, $a^{16}-b^{16}$, $2c-4c^3+2c^5$.
Printed answer:
- tabularl $\displaystyle \left( \frac{x}{2} + \frac{yz^2}{3m^3} \right) \left( \frac{x}{2} - \frac{yz^2}{3m^3} \right) $; $(x^2 - 3y)(x^4 + 3x^2y + 9 y^2) $; $(a^8 + b^8)(a^4 + b^4) $ etc.; $2c(c^2 - 1)^2$. tabular
verified: the printed answer passed a computed check
How it was checked
- factor: passes (x/2 + y*z**2/(3*m**3))*(x/2 - y*z**2/(3*m**3))
Exercise 54, problem 76b
Factor $\displaystyle \frac{x^2}{4}-\frac{y^2z^4}{9m^6}$, $x^6-27y^3$, $a^{16}-b^{16}$, $2c-4c^3+2c^5$.
Printed answer:
- tabularl $\displaystyle \left( \frac{x}{2} + \frac{yz^2}{3m^3} \right) \left( \frac{x}{2} - \frac{yz^2}{3m^3} \right) $; $(x^2 - 3y)(x^4 + 3x^2y + 9 y^2) $; $(a^8 + b^8)(a^4 + b^4) $ etc.; $2c(c^2 - 1)^2$. tabular
verified: the printed answer passed a computed check
How it was checked
- factor: passes (x**2 - 3*y)*(x**4 + 3*x**2*y + 9*y**2)
Exercise 54, problem 76c
Factor $\displaystyle \frac{x^2}{4}-\frac{y^2z^4}{9m^6}$, $x^6-27y^3$, $a^{16}-b^{16}$, $2c-4c^3+2c^5$.
Printed answer:
- tabularl $\displaystyle \left( \frac{x}{2} + \frac{yz^2}{3m^3} \right) \left( \frac{x}{2} - \frac{yz^2}{3m^3} \right) $; $(x^2 - 3y)(x^4 + 3x^2y + 9 y^2) $; $(a^8 + b^8)(a^4 + b^4) $ etc.; $2c(c^2 - 1)^2$. tabular
unverified: no computed check settled this one (yet)
How it was checked
- factor: the printed answer does not match the problem (a**8 + b**8)*(a**4 + b**4)
Exercise 54, problem 76d
Factor $\displaystyle \frac{x^2}{4}-\frac{y^2z^4}{9m^6}$, $x^6-27y^3$, $a^{16}-b^{16}$, $2c-4c^3+2c^5$.
Printed answer:
- tabularl $\displaystyle \left( \frac{x}{2} + \frac{yz^2}{3m^3} \right) \left( \frac{x}{2} - \frac{yz^2}{3m^3} \right) $; $(x^2 - 3y)(x^4 + 3x^2y + 9 y^2) $; $(a^8 + b^8)(a^4 + b^4) $ etc.; $2c(c^2 - 1)^2$. tabular
verified: the printed answer passed a computed check
How it was checked
- factor: passes 2*c*(c**2 - 1)**2
Exercise 54, problem 77
Extract the square root of $12x^4-24x+9+x^6-22x^3-4x^5+28x^2$.
Printed answer:
- $x^3 -2x^2 + 4x -3$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles x**3 - 2*x**2 + 4*x - 3
Exercise 54, problem 78
What must be subtracted from the sum of $4x^3+3x^2y-y^3$, $4x^2y-3x^3$, $7x^2y+9y^3-2x^2y$, to leave the remainder $2x^3-3x^2y+y^3$?
Printed answer:
- $7y^3 + 15x^2y - x^3$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 7*y**3 + 15*x**2*y - x**3
Exercise 54, problem 79
Find the G.C.F. of $x\left(x+1\right)^2$, $x^2(x^2-1)$, and $2x^3-2x^2-4x$.
Printed answer:
- $x(x + 1)$.
verified: the printed answer passed a computed check
How it was checked
- hcf: passes x*(x + 1)
Exercise 54, problem 8
I bought some five-cent stamps, and twice as many two-cent stamps, paying for the whole 81 cents. How many stamps of each kind did I buy?
Printed answer:
- $9$ fives; $18$ twos.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"f": "9", "t": "18"}
Exercise 54, problem 80
From one end of a line I cut off 5 feet less than one-fifth of it, and from the other end 4 feet more than one-fourth of it, and then there remained 34 feet. How long was the line?
Printed answer:
- 60 ft.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"L": "60"}
Exercise 54, problem 81
A can do twice as much work as B, B can do twice as much as C, and together they can complete a piece of work in 4 days. In what time can each alone complete the work.
Printed answer:
- A, 7 days; B, 14 days; C, 28 days.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"A": "7", "B": "14", "C": "28"}
Exercise 54, problem 82
Separate 57 into two parts, such that one divided by the other may give 5 as a quotient, with 3 as a remainder.
Printed answer:
- $9$; $48$.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the record may be misread {"x": "48", "y": "9"}
Exercise 54, problem 83
Divide 92 into two parts, such that one divided by the other may give 4 as a quotient, with 2 as a remainder.
Printed answer:
- $18$; $74$.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the record may be misread {"x": "74", "y": "18"}
Exercise 54, problem 84
Fourteen persons engaged a yacht, but before sailing, four of the company withdrew, by which the expense of each was increased $4. What was paid for the yacht?
Printed answer:
- $140.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"T": "140"}
Exercise 54, problem 85
Find two consecutive numbers such that a fifth of the larger shall equal the difference between a third and an eighth of the smaller.
Printed answer:
- $24$; $25$.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the record may be misread {"L": "25", "s": "24"}
Exercise 54, problem 86
A is 24 years older than B, and A’s age is as much above 50 as B’s is below 40. What is the age of each?
Printed answer:
- A, $57$ yrs.; B $33$ yrs.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the record may be misread {"A": "57", "B": "33"}
Exercise 54, problem 87
Find the number, whose double added to 16 will be as much above 70 as the number itself is below 60.
Printed answer:
- $38$.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the record may be misread {"x": "38"}
Exercise 54, problem 88
A hare takes 5 leaps to a dog’s 4, but 3 of the dog’s leaps are equal to 4 of the hare’s; the hare has a start of 20 leaps. How many leaps will the hare take before he is caught? *Suggestion*. Let $5x$ equal the number of leaps the hare will take, and let $m$ equal the length of one leap.
Printed answer:
- $300$ leaps.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the record may be misread {"n": "300"}
Exercise 54, problem 89
A greyhound takes 3 leaps to a hare’s 5, but 2 of the greyhound’s leaps are equal to 4 of the hare’s. If the hare has a start of 48 leaps, how soon will the greyhound overtake him?
Printed answer:
- With $144$ leaps.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the record may be misread {"n": "144"}
Exercise 54, problem 9
Three barns contain 58 tons of hay. In the first barn there are 3 tons more than in the second, and 7 less than in the third. How many tons in each barn?
Printed answer:
- $18$; $15$; $25$ tons.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {"a": "18", "b": "15", "c": "25"}
Exercise 54, problem 90
A hare has 40 leaps the start of a dog. When will he be caught if 5 of his leaps are equal to 4 of the dog’s, and if he takes 7 leaps while the dog takes 6?
Printed answer:
- After $560$ leaps.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the record may be misread {"n": "560"}