Fractional Equations

Excerpts

Fractional Equations
If a fraction is preceded by a **minus sign**, *the sign of every term of the numerator must be changed when the denominator is removed*.
Fractional Equations
Multiply each term by the L. C. M. of the denominators.
Fractional Equations
If the denominators contain both simple and compound expressions, it is generally best to remove the simple expressions first, and then the compound expressions. After each multiplication the result should be reduced to the simplest form.
Fractional Equations
Literal equations are equations in which some or all of the known numbers are represented by letters; the numbers regarded as known numbers are usually represented by the *first* letters of the alphabet.
Fractional Equations
By working *$2$ days each* they build $\frac{1}{12} + \frac{1}{15} + \frac{1}{20}$ of it.
Fractional Equations
Such an expression is called a **formula**, and the translation of this formula into words is called a **rule**.
Fractional Equations
A hare takes $4$ leaps to a greyhound’s $3$; but $2$ of the greyhound’s leaps are equivalent to $3$ of the hare’s.

Equations

Fractional Equations
i = prt
The simple interest equals the principal times the rate times the time.
Fractional Equations
p = \frac{i}{rt}
The principal is the interest divided by the product of the rate and the time.
Fractional Equations
p = \frac{a}{1 + rt}
The principal that amounts to a given sum equals the amount divided by one plus the product of rate and time.
Fractional Equations
t = \frac{a - p}{pr}
The time is the interest (amount minus principal) divided by the product of principal and rate.
Fractional Equations
r = \frac{a - p}{pt}
The rate is the interest (amount minus principal) divided by the product of principal and time.
Fractional Equations
x = \frac{s - d}{2}
The smaller of two numbers equals half the difference of their sum and their difference.
Fractional Equations
x = \frac{ab}{a + b}
Two agents working together need a time equal to the product of their individual times divided by their sum.

Problems

Exercise 51

Exercise 51, problem 1, p. 105
$\dfrac{x - 1}{2} = \dfrac{x + 1}{3}$.
Printed answer:
- $5$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [5]
Exercise 51, problem 10, p. 105
$\dfrac{4x}{x + 1} - \dfrac{x}{x - 2} = 3$.
Printed answer:
- $1$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [1]
Exercise 51, problem 11, p. 105
$\dfrac{2x + 1}{4} - \dfrac{4x - 1}{10} + 1 - \frac{1}{4} = 0$.
Printed answer:
- $-16$.
unverified: no computed check settled this one (yet)
How it was checked
- solve: the printed answer does not match the problem [-16]
Exercise 51, problem 12, p. 105
$\dfrac{x - 1}{5} - \dfrac{43 - 5x}{6} - \dfrac{3x - 1}{8} = 0$.
Printed answer:
- $11$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [11]
Exercise 51, problem 13, p. 105
$\dfrac{1}{x + 7} = \dfrac{2}{x + 1} - \dfrac{1}{x + 3}$.
Printed answer:
- $-4$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [-4]
Exercise 51, problem 14, p. 105
$\dfrac{1}{x + 4} + \dfrac{2}{x + 6} - \dfrac{3}{x + 5} = 0$.
Printed answer:
- $-2$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [-2]
Exercise 51, problem 15, p. 105
$\dfrac{4}{x^{2} - 1} + \dfrac{1}{x - 1} + \dfrac{1}{x + 1} = 0$.
Printed answer:
- $-2$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [-2]
Exercise 51, problem 16, p. 105
$\dfrac{3x + 1}{4} - \dfrac{5x - 4}{7} = 12 - 2x - \dfrac{x - 2}{3}$.
Printed answer:
- $5$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [5]
Exercise 51, problem 17, p. 105
$\frac{1}{8}(5x + 3) - \frac{1}{3}(3 - 4x) + \frac{1}{6}(9 - 5x) = \frac{1}{2}(31 - x)$.
Printed answer:
- $9$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [9]
Exercise 51, problem 18, p. 105
$\frac{1}{15} (34x - 56) - \frac{1}{5}(7x - 3) - \frac{1}{3}(7x - 5) = 0$.
Printed answer:
- $-1$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [-1]
Exercise 51, problem 2, p. 105
$\dfrac{3x - 1}{4} = \dfrac{2x + 1}{3}$.
Printed answer:
- $7$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [7]
Exercise 51, problem 3, p. 105
$\dfrac{6x - 19}{2} = \dfrac{2x - 11}{3}$.
Printed answer:
- $2\frac{1}{2}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [Rational(5, 2)]
Exercise 51, problem 4, p. 105
$\dfrac{7x - 40}{8} = \dfrac{9x - 80}{10}$.
Printed answer:
- $120$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [120]
Exercise 51, problem 5, p. 105
$\dfrac{3x - 116}{4} + \dfrac{180 - 5x}{6} = 0$.
Printed answer:
- $12$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [12]
Exercise 51, problem 6, p. 105
$\dfrac{3x - 4}{2} - \dfrac{3x - 1}{16} = \dfrac{6x - 5}{8}$.
Printed answer:
- $2\frac{1}{3}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [Rational(7, 3)]
Exercise 51, problem 7, p. 105
$\dfrac{x - 1}{8} - \dfrac{x + 1}{18} = 1$.
Printed answer:
- $17$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [17]
Exercise 51, problem 8, p. 105
$\dfrac{60 - x}{14} - \dfrac{3x - 5}{7} = \dfrac{3x}{4}$.
Printed answer:
- $4$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [4]
Exercise 51, problem 9, p. 105
$\dfrac{3x - 1}{11} - \dfrac{2 - x}{10} = \dfrac{6}{5}$.
Printed answer:
- $4$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [4]

Exercise 52

Exercise 52, problem 1, p. 106
$\frac{2}{3}(x + 1) - \frac{1}{7}(x + 5) = 1$.
Printed answer:
- $2$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 2
On the STU-32 (STU, rpn):
```
2 ENTER 3 ÷ 5 ENTER 7 ÷ −
```
```
1 x↔y −
```
```
2 ENTER 3 ÷ 1 ENTER 7 ÷ − ÷
```
Calculator: +2000000000000000000000000000000001E-33; the book prints 2. Run on the calculator core at firmware 628c96c.
Exercise 52, problem 10, p. 106
$\dfrac{5x + 3}{x - 1} + \dfrac{2x - 3}{2x - 1} = 6$.
Printed answer:
- $\frac{3}{7}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 3/7
Exercise 52, problem 11, p. 106
$\dfrac{3x}{4x + 1} + 1 = 2 - \dfrac{x}{2(2x - 1)}$.
Printed answer:
- $2$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 2
On the STU-32 (STU, rpn):
```
3 ENTER 2 × 1 − 4 − 2 x↔y ÷
```
Calculator: +2E+0; the book prints 2. Run on the calculator core at firmware 628c96c.
Exercise 52, problem 12, p. 106
$\dfrac{8x + 7}{5x + 4} - 1 = 1 - \dfrac{2x}{5x + 1}$.
Printed answer:
- $1$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 1
Exercise 52, problem 13, p. 106
$\dfrac{x + 1}{2(x - 1)} - \dfrac{x - 1}{x + 1} = \dfrac{17 - x^{2}}{2(x^{2} - 1)}$.
Printed answer:
- $3$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 3
Exercise 52, problem 2, p. 106
$\frac{6}{7}(x - 9) - \frac{1}{3}(5 - x) + 3x + 1 = 0$.
Printed answer:
- $2$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 2
On the STU-32 (STU, rpn):
```
6 ENTER 7 ÷ 9 +/− ×
```
```
5 ENTER 3 ÷ −
```
```
1 +
```
```
ENTER 6 ENTER 7 ÷ 1 ENTER 3 ÷ + 3 +
```
```
x↔y +/− x↔y ÷
```
Calculator: +2000000000000000000000000000000000E-33; the book prints 2. Run on the calculator core at firmware 628c96c.
Exercise 52, problem 3, p. 106
$\frac{1}{3}(5x - 24) + \frac{1}{7}(x - 2) - 2(x - 1) = 0$.
Printed answer:
- $-33$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes -33
On the STU-32 (STU, rpn):
```
24 ENTER 3 ÷ +/− 2 ENTER 7 ÷ − 2 +
```
```
5 ENTER 3 ÷ 1 ENTER 7 ÷ + 2 −
```
```
÷ +/−
```
Calculator: -3300000000000000000000000000000008E-32; the book prints -33. Run on the calculator core at firmware 628c96c.
Exercise 52, problem 4, p. 106
$\dfrac{x + 3}{4} + \dfrac{7x - 2}{5} = \dfrac{5x - 1}{4} + \dfrac{5x + 4}{9}$.
Printed answer:
- $1$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 1
On the STU-32 (STU, rpn):
```
1 ENTER 4 ÷ 5 ENTER 4 ÷ − 7 ENTER 5 ÷ + 5 ENTER 9 ÷ −
```
```
3 ENTER 4 ÷ 2 ENTER 5 ÷ − 1 ENTER 4 ÷ + 4 ENTER 9 ÷ −
```
```
x↔y ÷ +/−
```
Calculator: +1E+0; the book prints 1. Run on the calculator core at firmware 628c96c.
Exercise 52, problem 5, p. 106
$\dfrac{x + 1}{3} - \dfrac{x - 1}{4} = \dfrac{x - 2}{5} - \dfrac{x - 3}{6} + \dfrac{31}{60}$.
Printed answer:
- $\frac{2}{3}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 2/3
Exercise 52, problem 6, p. 106
$\dfrac{(2x - 1)(2 - x)}{2} + x^{2} - \dfrac{1 + 3x}{2} = 0$.
Printed answer:
- $1\frac{1}{2}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 3/2
Exercise 52, problem 7, p. 106
$\dfrac{6x - 11}{4} - \dfrac{3 - 4x}{6} = \dfrac{4}{3} - \dfrac{x}{8}$.
Printed answer:
- $2$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 2
On the STU-32 (STU, rpn):
```
4 ENTER 3 ÷ 11 ENTER 4 ÷ + 3 ENTER 6 ÷ +
```
```
6 ENTER 4 ÷ 4 ENTER 6 ÷ + 1 ENTER 8 ÷ + ÷
```
Calculator: +2000000000000000000000000000000000E-33; the book prints 2. Run on the calculator core at firmware 628c96c.
Exercise 52, problem 8, p. 106
$\dfrac{x + 6}{4} - \dfrac{16 - 3x}{12} = 4\frac{1}{6}$.
Printed answer:
- $8$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 8
On the STU-32 (STU, rpn):
```
6 ENTER 4 ÷ 16 ENTER 12 ÷ − +/−
```
```
4 ENTER 1 ENTER 6 ÷ + +
```
```
1 ENTER 4 ÷ 3 ENTER 12 ÷ + ÷
```
Calculator: +80000000000000000000000000000000E-31; the book prints 8. Run on the calculator core at firmware 628c96c.
Exercise 52, problem 9, p. 106
$x - \dfrac{x - 2}{3} = \dfrac{x + 23}{4} - \dfrac{10 + x}{5}$.
Printed answer:
- $5$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 5
On the STU-32 (STU, rpn):
```
2 ENTER 3 ÷
```
```
23 ENTER 4 ÷ −
```
```
10 ENTER 5 ÷ +
```
```
1 ENTER 3 ÷
```
```
1 x↔y −
```
```
1 ENTER 4 ÷ −
```
```
1 ENTER 5 ÷ +
```
```
+/− ÷
```
Calculator: +4999999999999999999999999999999999E-33; the book prints 5. Run on the calculator core at firmware 628c96c.

Exercise 53

Exercise 53, problem 1, p. 107
$\dfrac{10x + 13}{18} - \dfrac{x + 2}{x - 3} = \dfrac{5x - 4}{9}$.
Printed answer:
- $33$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 33
On the STU-32 (STU, rpn):
```
13 ENTER 3 ×
```
```
18 ENTER 18 × 9 ÷ +
```
```
4 ENTER 18 × 3 × 9 ÷ +
```
```
3 ÷
```
Calculator: +33E+0; the book prints 33. Run on the calculator core at firmware 628c96c.
Exercise 53, problem 2, p. 107
$\dfrac{6x + 7}{10} - \dfrac{3x + 1}{5} = \dfrac{x - 1}{3x - 4}$.
Printed answer:
- $2$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 2
Exercise 53, problem 3, p. 107
$\dfrac{11x - 12}{14} - \dfrac{11x - 7}{19x + 7} = \dfrac{22x - 36}{28}$.
Printed answer:
- $3\frac{1}{2}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 7/2
On the STU-32 (STU, rpn):
```
28 ENTER 14 ÷
```
```
12 +/− × 7 ×
```
```
28 ENTER 7 × +
```
```
36 ENTER 7 × +
```
```
36 ENTER 19 ×
```
```
28 ENTER 14 ÷ 12 × 19 × −
```
```
28 ENTER 11 × −
```
```
÷ +/−
```
Calculator: +35E-1; the book prints 7/2. Run on the calculator core at firmware 628c96c.
Exercise 53, problem 4, p. 107
$\dfrac{2x - 1}{5} + \dfrac{2x - 3}{17x - 12} = \dfrac{4x - 3}{10}$.
Printed answer:
- $1\frac{5}{37}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 42/37
Exercise 53, problem 5, p. 107
$\dfrac{11x - 13}{7} - \dfrac{13x + 7}{3x + 7} = \dfrac{22x - 75}{14}$.
Printed answer:
- $7$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 7
Exercise 53, problem 6, p. 107
$\dfrac{6x - 13}{2x + 3} + \dfrac{6x + 7}{9} - \dfrac{2x + 4}{3} = 0$.
Printed answer:
- $3$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 3

Exercise 54

Exercise 54, problem 1, p. 108
$a(x - a) = b(x - b)$.
Printed answer:
- $a + b$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes a + b
Exercise 54, problem 10, p. 108
$(a + bx)(c + d) = (a + b)(c + dx)$.
Printed answer:
- $1$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 1
Exercise 54, problem 11, p. 108
$\dfrac{x}{a - b} - \dfrac{3a}{a + b} = \dfrac{bx}{a^{2} - b^{2}}$.
Printed answer:
- $3(a - b)$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 3*(a - b)
Exercise 54, problem 2, p. 108
$(a + b)x + (a - b)x = a^{2}$.
Printed answer:
- $\dfrac{a}{2}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes a/2
Exercise 54, problem 3, p. 108
$(a + b)x - (a - b)x = b^{2}$.
Printed answer:
- $\dfrac{b}{2}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes b/2
Exercise 54, problem 4, p. 108
$(2x - a) + (x - 2a) = 3a$.
Printed answer:
- $2a$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 2*a
Exercise 54, problem 5, p. 108
$(x + a + b) + (x + a - b) = 2b$.
Printed answer:
- $b - a$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes b - a
Exercise 54, problem 6, p. 108
$(x - a)(x - b) = x(x + c)$.
Printed answer:
- $\dfrac{ab}{a + b + c}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes a*b/(a + b + c)
Exercise 54, problem 7, p. 108
$x^{2} + b^{2} = (a - x)(a - x)$.
Printed answer:
- $\dfrac{a^{2} - b^{2}}{2a}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes (a**2 - b**2)/(2*a)
Exercise 54, problem 8, p. 108
$(a + b)(2 - x) = (a - b)(2 + x)$.
Printed answer:
- $\dfrac{2b}{a}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 2*b/a
Exercise 54, problem 9, p. 108
$(x - a)(2x - a) = 2(x - b)^{2}$.
Printed answer:
- $\dfrac{2b^{2} - a^{2}}{4b - 3a}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes (2*b**2 - a**2)/(4*b - 3*a)

Exercise 55

Exercise 55, problem 1, p. 109
The difference between the fifth and seventh parts of a certain number is $2$. Find the number.
Printed answer:
- $35$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {x: 35}
On the STU-32 (STU, rpn):
```
2 ENTER 5 × 7 ×
```
```
7 ENTER 5 − ÷
```
Calculator: +35E+0; the book prints 35. Run on the calculator core at firmware 628c96c.
Exercise 55, problem 2, p. 109
One-half of a certain number exceeds the sum of its fifth and seventh parts by $11$. Find the number.
Printed answer:
- $70$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {x: 70}
On the STU-32 (STU, rpn):
```
11 ENTER
```
```
2 1/x
```
```
5 1/x −
```
```
7 1/x −
```
```
÷
```
Calculator: +7000000000000000000000000000000002E-32; the book prints 70. Run on the calculator core at firmware 628c96c.
Exercise 55, problem 3, p. 109
The sum of the third and sixth parts of a certain number exceeds the difference of its sixth and ninth parts by $16$. Find the number.
Printed answer:
- $36$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {x: 36}
On the STU-32 (STU, rpn):
```
16 ENTER 3 1/x 9 1/x +
```
```
÷
```
Calculator: +3600000000000000000000000000000000E-32; the book prints 36. Run on the calculator core at firmware 628c96c.
Exercise 55, problem 4, p. 109
There are two consecutive numbers, $x$ and $x + 1$, such that one-half the larger exceeds one-third the smaller number by $10$. Find the numbers.
Printed answer:
- $57$, $58$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {x: 57}
On the STU-32 (STU, rpn):
```
10 ENTER 1 ENTER 2 ÷ −
```
```
2 ENTER 3 × ×
```
Calculator: +570E-1; the book prints 57. Run on the calculator core at firmware 628c96c.

Exercise 56

Exercise 56, problem 1, p. 110
The sum of two numbers is $100$, and if the greater is divided by the smaller number, the quotient is $4$ and the remainder $5$. Find the numbers.
Printed answer:
- $81$, $19$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {x: 81, y: 19}
Exercise 56, problem 2, p. 110
The sum of two numbers is $124$, and if the greater is divided by the smaller number, the quotient is $4$ and the remainder $4$. Find the numbers.
Printed answer:
- $100$, $24$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {x: 100, y: 24}
Exercise 56, problem 3, p. 110
The difference of two numbers is $49$, and if the greater is divided by the smaller, the quotient is $4$ and the remainder $4$. Find the numbers.
Printed answer:
- $64$, $15$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {x: 64, y: 15}
Exercise 56, problem 4, p. 110
The difference of two numbers is $91$, and if the greater is divided by the smaller, the quotient is $8$ and the remainder $7$. Find the numbers.
Printed answer:
- $103$, $12$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {x: 103, y: 12}
Exercise 56, problem 5, p. 110
Divide $320$ into two parts such that the smaller part is contained in the larger part $11$ times, with a remainder of $20$.
Printed answer:
- $295$, $25$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {x: 295, y: 25}
Exercise 56, problem None, p. 110
The sum of two numbers is $63$, and if the greater is divided by the smaller number, the quotient is $2$ and the remainder $3$. Find the numbers.
Printed answer:
- x &= 43.
verified: the printed answer passed a computed check
How it was checked
- solve: passes 43
On the STU-32 (STU, rpn):
```
63 ENTER 2 × 3 +
```
```
1 ENTER 2 + ÷
```
Calculator: +43E+0; the book prints 43. Run on the calculator core at firmware 628c96c.

Exercise 57

Exercise 57, problem 1, p. 111
A son is one-fourth as old as his father. In $24$ years he will be one-half as old. Find the age of the son.
Printed answer:
- $12$ yr.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {s: 12}
On the STU-32 (STU, rpn):
```
24 ENTER 2 ÷
```
Calculator: +12E+0; the book prints 12. Run on the calculator core at firmware 628c96c.
Exercise 57, problem 2, p. 111
B’s age is one-sixth of A’s age. In $15$ years B’s age will be one-third of A’s age. Find their ages.
Printed answer:
- A, $60$ yr.; B, $10$ yr.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {A: 60, B: 10}
Exercise 57, problem 3, p. 111
The sum of the ages of A and B is $30$ years, and $5$ years hence B’s age will be one-third of A’s. Find their ages.
Printed answer:
- A, $25$ yr.; B, $5$ yr.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {A: 25, B: 5}
Exercise 57, problem 4, p. 111
A father is $35$ years old, and his son is one-fourth of that age. In how many years will the son be half as old as his father?
Printed answer:
- $17\frac{1}{2}$ yr.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {x: Rational(35, 2)}
On the STU-32 (STU, rpn):
```
35 ENTER 35 ENTER 4 ÷
```
```
2 × −
```
Calculator: +1750E-2; the book prints 35/2. Run on the calculator core at firmware 628c96c.
Exercise 57, problem 5, p. 111
A is $60$ years old, and B’s age is two-thirds of A’s. How many years ago was B’s age one-fifth of A’s?
Printed answer:
- $35$ yr.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {k: 35}
On the STU-32 (STU, rpn):
```
60 ENTER 2 × 3 ÷
```
```
5 ×
```
```
60 −
```
```
4 ÷
```
Calculator: +35E+0; the book prints 35. Run on the calculator core at firmware 628c96c.
Exercise 57, problem 6, p. 111
A son is one-third as old as his father. Four years ago he was only one-fourth as old as his father. What is the age of each?
Printed answer:
- Son, $12$ yr.; father, $36$ yr.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {S: 12, F: 36}
Exercise 57, problem 7, p. 111
A is $50$ years old, and B is half as old as A. In how many years will B be two-thirds as old as A?
Printed answer:
- $25$ yr.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {y: 25}
On the STU-32 (STU, rpn):
```
50 ENTER 2 ×
```
```
50 ENTER 2 ÷
```
```
3 ×
```
```
−
```
Calculator: +25E+0; the book prints 25. Run on the calculator core at firmware 628c96c.
Exercise 57, problem 8, p. 111
B is one-half as old as A. Ten years ago he was one-fourth as old as A. What are their present ages?
Printed answer:
- A, $30$ yr.; B, $15$ yr.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {A: 30, B: 15}
Exercise 57, problem 9, p. 111
The sum of the ages of a father and his son is $80$ years. The son’s age increased by $5$ years is one-fourth of the father’s age. Find their ages.
Printed answer:
- Son, $12$ yr.; father, $68$ yr.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {S: 12, F: 68}
Exercise 57, problem None, p. 111
Eight years ago a boy was one-fourth as old as he will be one year hence. How old is he now?
Printed answer:
- (none printed)
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation 11
On the STU-32 (STU, rpn):
```
8 ENTER 4 × 1 +
```
```
4 ENTER 1 − ÷
```
Calculator: +11E+0; the book prints 11. Run on the calculator core at firmware 628c96c.

Exercise 58

Exercise 58, problem 1, p. 112
A can do a piece of work in $3$ days, B in $5$ days, and C in $6$ days. How long will it take them to do it working together?
Printed answer:
- $1\frac{3}{7}$ dy.
verified: the printed answer passed a computed check
How it was checked
- solve: passes Rational(10,7)
On the STU-32 (STU, rpn):
```
3 1/x 5 1/x + 6 1/x + 1/x
```
Calculator: +1428571428571428571428571428571429E-33; the book prints 10/7. Run on the calculator core at firmware 628c96c.
Exercise 58, problem 2, p. 112
A can do a piece of work in $5$ days, B in $4$ days, and C in $3$ days. How long will it take them together to do the work?
Printed answer:
- $1\frac{13}{47}$ dy.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation Rational(60,47)
On the STU-32 (STU, rpn):
```
5 1/x
```
```
4 1/x +
```
```
3 1/x +
```
```
1/x
```
Calculator: +1276595744680851063829787234042553E-33; the book prints 60/47. Run on the calculator core at firmware 628c96c.
Exercise 58, problem 3, p. 112
A can do a piece of work in $2\frac{1}{2}$ days, B in $3\frac{1}{2}$ days, and C in $3\frac{3}{4}$ days. How long will it take them together to do the work?
Printed answer:
- $1\frac{1}{20}$ dy.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation Rational(21,20)
On the STU-32 (STU, rpn):
```
2 ENTER 1 ENTER 2 ÷ + 1/x
```
```
3 ENTER 1 ENTER 2 ÷ + 1/x +
```
```
3 ENTER 3 ENTER 4 ÷ + 1/x +
```
```
1/x
```
Calculator: +1050000000000000000000000000000000E-33; the book prints 21/20. Run on the calculator core at firmware 628c96c.
Exercise 58, problem 4, p. 112
A can do a piece of work in $10$ days, B in $12$ days; A and B together, with the help of C, can do the work in $4$ days. How long will it take C alone to do the work?
Printed answer:
- $15$ dy.
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):
```
1 ENTER 4 ÷ 1 ENTER 10 ÷ −
```
```
1 ENTER 12 ÷ − 1/x
```
Calculator: +1500000000000000000000000000000000E-32; the book prints 15. Run on the calculator core at firmware 628c96c.
Exercise 58, problem 5, p. 112
A and B together can mow a field in $10$ hours, A and C in $12$ hours, and A alone in $20$ hours. In what time can B and C together mow the field?
Printed answer:
- $12$ hr.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {a: Rational(1,20), b: Rational(1,20), c: Rational(1,30), t: 12}
Exercise 58, problem 6, p. 112
A and B together can build a wall in $12$ days, A and C in $15$ days, B and C in $20$ days. In what time can they build the wall if they all work together?
Printed answer:
- $10$ dy.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {a: Rational(1,20), b: Rational(1,30), c: Rational(1,60), t: 10}

Exercise 59

Exercise 59, problem 1, p. 113
A cistern can be filled by three pipes in $16$, $24$, and $32$ hours, respectively. In what time will it be filled by all the pipes together?
Printed answer:
- $7\frac{5}{13}$ hr.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {x: 96/13}
Exercise 59, problem 2, p. 113
A tank can be filled by two pipes in $3$ hours and $4$ hours, respectively, and can be emptied by a third pipe in $6$ hours. In what time will the cistern be filled if the pipes are all running together?
Printed answer:
- $2\frac{2}{5}$ hr.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {x: 12/5}
On the STU-32 (STU, rpn):
```
3 1/x 4 1/x +
```
```
6 1/x −
```
```
1/x
```
Calculator: +2400000000000000000000000000000000E-33; the book prints 12/5. Run on the calculator core at firmware 628c96c.
Exercise 59, problem 3, p. 113
A tank can be filled by three pipes in $1$ hour and $40$ minutes, $3$ hours and $20$ minutes, and $5$ hours, respectively. In what time will the tank be filled if all three pipes are running together?
Printed answer:
- $\frac{10}{11}$ hr.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {x: 10/11}
On the STU-32 (STU, rpn):
```
3 ENTER 5 ÷
```
```
3 ENTER 10 ÷
```
```
+
```
```
1 ENTER 5 ÷
```
```
+
```
```
1/x
```
Calculator: +9090909090909090909090909090909091E-34; the book prints 10/11. Run on the calculator core at firmware 628c96c.
Exercise 59, problem 4, p. 113
A cistern can be filled by three pipes in $2\frac{1}{3}$ hours, $3\frac{1}{2}$ hours, and $4\frac{2}{3}$ hours, respectively. In what time will the cistern be filled if all the pipes are running together?
Printed answer:
- $1\frac{1}{13}$ hr.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {x: 14/13}
On the STU-32 (STU, rpn):
```
1 ENTER 3 ÷ 2 + 1/x
```
```
1 ENTER 2 ÷ 3 + 1/x +
```
```
2 ENTER 3 ÷ 4 + 1/x +
```
```
1/x
```
Calculator: +1076923076923076923076923076923077E-33; the book prints 14/13. Run on the calculator core at firmware 628c96c.
Exercise 59, problem 5, p. 113
A cistern has three pipes. The first pipe will fill the cistern in $12$ hours, the second in $20$ hours, and all three pipes together will fill it in $6$ hours. How long will it take the third pipe alone to fill it?
Printed answer:
- $30$ hr.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {x: 30}
On the STU-32 (STU, rpn):
```
6 1/x
```
```
12 1/x −
```
```
20 1/x −
```
```
1/x
```
Calculator: +2999999999999999999999999999999997E-32; the book prints 30. Run on the calculator core at firmware 628c96c.

Exercise 60

Exercise 60, problem 1, p. 114
A sets out from Boston and walks towards Portland at the rate of $3$ miles an hour. Three hours afterward B sets out from the same place and walks in the same direction at the rate of $4$ miles an hour. How far from Boston will B overtake A?
Printed answer:
- $36$ mi.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {x: 36}
On the STU-32 (STU, rpn):
```
3 ENTER 3 ×
```
```
4 ×
```
```
4 ENTER 3 −
```
```
÷
```
Calculator: +36E+0; the book prints 36. Run on the calculator core at firmware 628c96c.
Exercise 60, problem 2, p. 114
A courier who goes at the rate of $6\frac{1}{2}$ miles an hour is followed, after $4$ hours, by another who goes at the rate of $7\frac{1}{2}$ miles an hour. In how many hours will the second overtake the first?
Printed answer:
- $26$ hr.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {x: 26}
On the STU-32 (STU, rpn):
```
7 ENTER 1 ENTER 2 ÷ + 6 ENTER 1 ENTER 2 ÷ + −
```
```
6 ENTER 1 ENTER 2 ÷ + 4 × x↔y ÷
```
Calculator: +26E+0; the book prints 26. Run on the calculator core at firmware 628c96c.
Exercise 60, problem 3, p. 114
A person walks to the top of a mountain at the rate of two miles an hour, and down the same way at the rate of $4$ miles an hour. If he is out $6$ hours, how far is it to the top of the mountain?
Printed answer:
- $8$ mi.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {x: 8}
On the STU-32 (STU, rpn):
```
6 ENTER 2 1/x 4 1/x + ÷
```
Calculator: +8E+0; the book prints 8. Run on the calculator core at firmware 628c96c.
Exercise 60, problem 4, p. 114
In going a certain distance, a train travelling at the rate of $40$ miles an hour takes $2$ hours less than a train travelling $30$ miles an hour. Find the distance.
Printed answer:
- $240$ mi.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {d: 240}
On the STU-32 (STU, rpn):
```
2 ENTER 30 1/x
```
```
40 1/x −
```
```
÷
```
Calculator: +2400000000000000000000000000000001E-31; the book prints 240. Run on the calculator core at firmware 628c96c.
Exercise 60, problem None, p. 114
A courier who travels $6$ miles an hour is followed, after $2$ hours, by a second courier who travels $7\frac{1}{2}$ miles an hour. In how many hours will the second courier overtake the first?
Printed answer:
- (none printed)
verified: the printed answer passed a computed check
How it was checked
- solve: passes 10
On the STU-32 (STU, rpn):
```
7 ENTER 1 ENTER 2 ÷ +
```
```
ENTER ENTER 6 −
```
```
x↔y 2 ×
```
```
x↔y ÷
```
Calculator: +10E+0; the book prints 10. Run on the calculator core at firmware 628c96c.

Exercise 61

Exercise 61, problem 1, p. 115
A hound makes $3$ leaps while a rabbit makes $5$; but $1$ of the hound’s leaps is equivalent to $2$ of the rabbit’s. The rabbit has a start of $120$ leaps. How many leaps will the rabbit take before she is caught?
Printed answer:
- $600$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {R: 600}
On the STU-32 (STU, rpn):
```
120 ENTER 5 ×
```
```
2 ENTER 3 × 5 − ÷
```
Calculator: +600E+0; the book prints 600. Run on the calculator core at firmware 628c96c.
Exercise 61, problem 2, p. 115
A rabbit takes $6$ leaps to a dog’s $5$, and $7$ of the dog’s leaps are equivalent to $9$ of the rabbit’s. The rabbit has a start of $60$ of her own leaps. How many leaps must the dog take to catch the rabbit?
Printed answer:
- $700$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {D: 700}
On the STU-32 (STU, rpn):
```
60 ENTER 9 ENTER 7 ÷ 6 ENTER 5 ÷ − ÷
```
Calculator: +6999999999999999999999999999999977E-31; the book prints 700. Run on the calculator core at firmware 628c96c.
Exercise 61, problem 3, p. 115
A dog makes $4$ leaps while a rabbit makes $5$; but $3$ of the dog’s leaps are equivalent to $4$ of the rabbit’s. The rabbit has a start of $90$ of the *dog’s leaps*. How many leaps will each take before the rabbit is caught?
Printed answer:
- Dog, $1440$; rabbit, $1800$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {D: 1440, R: 1800}

Exercise 62

Exercise 62, problem 1, p. 116
Find the time between $5$ and $6$ o’clock when the hands of a clock are together.
Printed answer:
- $27\frac{3}{11}$ min. past 5 o’clock.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {x: Rational(300, 11)}
On the STU-32 (STU, rpn):
```
25 ENTER 12 ×
```
```
12 ENTER 1 − ÷
```
Calculator: +2727272727272727272727272727272727E-32; the book prints 300/11. Run on the calculator core at firmware 628c96c.
Exercise 62, problem 2, p. 116
Find the time between $2$ and $3$ o’clock when the hands of a clock are at right angles to each other.
Printed answer:
- $27\frac{3}{11}$ min. past 2 o’clock.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {x: Rational(300, 11)}
On the STU-32 (STU, rpn):
```
25 ENTER 1 ENTER 12 1/x −
```
```
÷
```
Calculator: +2727272727272727272727272727272727E-32; the book prints 300/11. Run on the calculator core at firmware 628c96c.
Exercise 62, problem 3, p. 116
Find the time between $2$ and $3$ o’clock when the hands of a clock point in opposite directions.
Printed answer:
- $43\frac{7}{11}$ min. past 2 o’clock.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {x: Rational(480, 11)}
Exercise 62, problem 4, p. 116
Find the time between $1$ and $2$ o’clock when the hands of a clock are at right angles to each other.
Printed answer:
- $21\frac{9}{11}$ min. past 1 o’clock.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {x: Rational(240, 11)}
On the STU-32 (STU, rpn):
```
20 ENTER 1 ENTER 12 1/x −
```
```
÷
```
Calculator: +2181818181818181818181818181818182E-32; the book prints 240/11. Run on the calculator core at firmware 628c96c.
Exercise 62, problem 5, p. 116
Find the time between $1$ and $2$ o’clock when the hands of a clock point in opposite directions.
Printed answer:
- $38\frac{2}{11}$ min. past 1 o’clock.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {x: Rational(420, 11)}
On the STU-32 (STU, rpn):
```
35 ENTER 1 ENTER 12 1/x − ÷
```
Calculator: +3818181818181818181818181818181818E-32; the book prints 420/11. Run on the calculator core at firmware 628c96c.
Exercise 62, problem 6, p. 116
At what time between $7$ and $8$ o’clock are the hands of a watch together?
Printed answer:
- $38\frac{2}{11}$ min. past 7 o’clock.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {x: Rational(420, 11)}
On the STU-32 (STU, rpn):
```
35 ENTER 12 × 12 ENTER 12 ÷ 12 − +/− ÷
```
Calculator: +3818181818181818181818181818181818E-32; the book prints 420/11. Run on the calculator core at firmware 628c96c.

Exercise 63

Exercise 63, problem 1, p. 117
A rectangle has its length and breadth respectively $7$ feet longer and $6$ feet shorter than the side of the equivalent square. Find its area.
Printed answer:
- $1764$ sq. ft.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {x: 42, A: 1764}
Exercise 63, problem 2, p. 117
The length of a floor exceeds the breadth by $5$ feet. If each dimension were $1$ foot more, the area of the floor would be $42$ sq. ft. more. Find its dimensions.
Printed answer:
- $18$ ft. by $23$ ft.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {B: 18, L: 23}
Exercise 63, problem 3, p. 117
A rectangle whose length is $6$ feet more than its breadth, would have its area $35$ sq. ft. more, if each dimension were $1$ foot more. Find its dimensions.
Printed answer:
- $14$ ft. by $20$ ft.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {B: 14, L: 20}
Exercise 63, problem 4, p. 117
The length of a rectangle exceeds its width by $3$ feet. If the length is increased by $3$ feet and the width diminished by $2$ feet, the area will not be altered. Find its dimensions.
Printed answer:
- $12$ ft. by $15$ ft.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {W: 12, L: 15}
Exercise 63, problem 5, p. 117
The length of a floor exceeds its width by $10$ feet If each dimension were $2$ feet more, the area would be $144$ sq. ft. more. Find its dimensions.
Printed answer:
- $30$ ft. by $40$ ft.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {W: 30, L: 40}

Exercise 64

Exercise 64, problem 1, p. 121
The sum of two angles is $120°\, 30'\, 30''$ and their difference $59°\, 30'\, 30''$. Find the angles.
Printed answer:
- $90°\,0'\,30''$; $30°\,30'$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {x: Rational(324030, 3600), y: Rational(61, 2)}
Exercise 64, problem 10, p. 121
Find the time required for $$160$ to amount to $$250$ at $6$%.
Printed answer:
- $9\frac{3}{8}$ yr.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {t: Rational(75, 8)}
On the STU-32 (STU, rpn):
```
250 ENTER 160 −
```
```
160 ENTER 6 × 100 ÷
```
```
÷
```
Calculator: +9375E-3; the book prints 75/8. Run on the calculator core at firmware 628c96c.
Exercise 64, problem 11, p. 121
How much money must be invested at $5$% to yield an annual income of $$1250$?
Printed answer:
- $$25,000$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {P: 25000}
On the STU-32 (STU, rpn):
```
1250 ENTER 5 ENTER 100 ÷ ÷
```
Calculator: +250E+2; the book prints 25000. Run on the calculator core at firmware 628c96c.
Exercise 64, problem 12, p. 121
Find the principal that will produce $$100$ a month if invested at $6$% per annum.
Printed answer:
- $$20,000$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {P: 20000}
On the STU-32 (STU, rpn):
```
100 ENTER 12 ×
```
```
100 ×
```
```
6 ÷
```
Calculator: +20000E+0; the book prints 20000. Run on the calculator core at firmware 628c96c.
Exercise 64, problem 13, p. 121
Find the rate if the interest on $$1000$ for $8$ months is $$40$.
Printed answer:
- $$6$%.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {r: Rational(3, 50)}
Exercise 64, problem 14, p. 121
Find the time for a sum of money on interest at $5$% to double itself.
Printed answer:
- $20$ yr.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {t: 20}
On the STU-32 (STU, rpn):
```
5 ENTER 100 ÷
```
```
1/x
```
Calculator: +2E+1; the book prints 20. Run on the calculator core at firmware 628c96c.
Exercise 64, problem 2, p. 121
Find the interest of $$1000$ for $3$ years and $4$ months at $4$%.
Printed answer:
- $$133\frac{1}{3}$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes Rational(400, 3)
Exercise 64, problem 3, p. 121
Find the principal that will amount to $$2280$ in $3$ years and $6$ months at $4$%.
Printed answer:
- $$2000$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {P: 2000}
On the STU-32 (STU, rpn):
```
6 ENTER 12 ÷ 3 +
```
```
4 ENTER 100 ÷ ×
```
```
1 +
```
```
2280 x↔y ÷
```
Calculator: +2E+3; the book prints 2000. Run on the calculator core at firmware 628c96c.
Exercise 64, problem 4, p. 121
Find the principal that will produce $$280$ interest in $2$ years and $4$ months at $3$%.
Printed answer:
- $$4000$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {P: 4000}
On the STU-32 (STU, rpn):
```
4 ENTER 12 ÷ 2 +
```
```
3 × 100 ÷
```
```
280 x↔y ÷
```
Calculator: +4000000000000000000000000000000001E-30; the book prints 4000. Run on the calculator core at firmware 628c96c.
Exercise 64, problem 5, p. 121
Find the principal that will produce $$270$ interest in $1$ year and $6$ months at $6$%.
Printed answer:
- $$3000$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {P: 3000}
On the STU-32 (STU, rpn):
```
270 ENTER 6 ENTER 100 ÷ ÷
```
```
3 ENTER 2 ÷ ÷
```
Calculator: +3E+3; the book prints 3000. Run on the calculator core at firmware 628c96c.
Exercise 64, problem 6, p. 121
Find the principal that will amount to $$590$ in $4$ years at $4\frac{1}{2}$%.
Printed answer:
- $$500$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {P: 500}
On the STU-32 (STU, rpn):
```
4 ENTER 1 ENTER 2 ÷ +
```
```
100 ÷
```
```
4 ×
```
```
1 +
```
```
590 x↔y ÷
```
Calculator: +5E+2; the book prints 500. Run on the calculator core at firmware 628c96c.
Exercise 64, problem 7, p. 121
Find the rate if the amount of $$250$ for $4$ years is $$300$.
Printed answer:
- $5$%.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {r: Rational(1, 20)}
On the STU-32 (STU, rpn):
```
300 ENTER 250 ÷
```
```
1 −
```
```
4 ÷
```
Calculator: +5E-2; the book prints 1/20. Run on the calculator core at firmware 628c96c.
Exercise 64, problem 8, p. 121
Find the rate if $$1000$ amounts to $$2000$ in $16$ years and $8$ months.
Printed answer:
- $6$%.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {r: Rational(3, 50)}
Exercise 64, problem 9, p. 121
Find the time required for the interest on $$400$ to be $$54$ at $4\frac{1}{2}$%.
Printed answer:
- $3$ yr.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {t: 3}
On the STU-32 (STU, rpn):
```
4 ENTER 1 ENTER 2 ÷ +
```
```
100 ÷
```
```
400 ×
```
```
54 x↔y ÷
```
Calculator: +3E+0; the book prints 3. Run on the calculator core at firmware 628c96c.