MODES OF REPRESENTING THE OPERATIONS

Excerpts

MODES OF REPRESENTING THE OPERATIONS
The sum of $y$ + $y$ + $y$ + etc. written seven times is 7$y$.
MODES OF REPRESENTING THE OPERATIONS
The 7 and $x$ are called the coefficients of the number following.
MODES OF REPRESENTING THE OPERATIONS
We find a difficulty in solving this last example, because we do not know just how large $x, b, m, y$, and $z$ are with reference to each other.
MODES OF REPRESENTING THE OPERATIONS
In algebra, therefore, numbers are considered as increasing from zero in opposite directions. Those in one direction are called Positive Numbers (or + numbers); those in the other direction Negative Numbers (or - numbers).
MODES OF REPRESENTING THE OPERATIONS
Negative numbers are usually spoken of as less than zero, because they are used to represent losses. To illustrate: suppose a man’s money affairs be such that his debts just equal his assets, we say that he is worth nothing.

Equations

MODES OF REPRESENTING THE OPERATIONS
m - y
A farmer who sold a sheep for m dollars and gained y dollars paid m - y dollars for it.
MODES OF REPRESENTING THE OPERATIONS
x - b + m + y - z
Taking east as positive and combining the separate eastward and westward distances gives the man's net distance from his starting point as x - b + m + y - z.
MODES OF REPRESENTING THE OPERATIONS
x \cdot x = xx = x^2
The product of x with itself, written xx, is the second power of x, written x^2.
MODES OF REPRESENTING THE OPERATIONS
x \cdot x \cdot x = xxx = x^3
The product of three factors each equal to x, written xxx, is the third power of x, written x^3.

Problems

Exercise 10

Exercise 10, problem 1
Charles walked $x$ miles and rode 9 miles. How far did he go?
Printed answer:
- $x + 9$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles x + 9
Exercise 10, problem 10
If $d$ is a whole number, what is the next larger number?
Printed answer:
- $d + 1$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles d + 1
Exercise 10, problem 11
A boy bought a pound of butter for $y$ cents, a pound of meat for $z$ cents, and a bunch of lettuce for $s$ cents. How much did they all cost?
Printed answer:
- $y + z + s$ cts.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles y + z + s
Exercise 10, problem 12
What is the next whole number larger than $m$?
Printed answer:
- $m + 1$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles m + 1
Exercise 10, problem 13
What is the sum of $x$ taken $y$ times?
Printed answer:
- $yx$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles x*y
Exercise 10, problem 14
A merchant sold $x$ barrels of flour one week, 40 the next week, and $a$ barrels the following week. How many barrels did he sell?
Printed answer:
- $x + 40 + a$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles x + 40 + a
Exercise 10, problem 15
Find two numbers whose sum is 74 and whose difference is 18.
Printed answer:
- $28$; $46$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes {x: 46, y: 28}
Exercise 10, problem 2
A merchant bought $a$ barrels of sugar and $p$ barrels of molasses. How many barrels in all did he buy?
Printed answer:
- $a + p$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles a + p
Exercise 10, problem 3
What is the sum of $b$ + $b$ + $b$ + etc. written eight times?
Printed answer:
- $8b$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 8*b
Exercise 10, problem 4
Express the sum of $x$ and $y$.
Printed answer:
- $x + y$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles x + y
Exercise 10, problem 5
There are $c$ boys at play, and 5 others join them. How many boys are there in all?
Printed answer:
- $c + 5$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles c + 5
Exercise 10, problem 6
What is the sum of $x$ + $x$ + $x$ + etc. written $d$ times?
Printed answer:
- $dx$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles d*x
Exercise 10, problem 7
A lady bought a silk dress for $m$ dollars, a muff for $l$ dollars, a shawl for $v$ dollars, and a pair of gloves for $c$ dollars. What was the entire cost?
Printed answer:
- $m + l + v + c$ dols.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles m + l + v + c
Exercise 10, problem 8
George is $x$ years old, Martin is $y$, and Morgan is $z$ years. What is the sum of their ages?
Printed answer:
- $x + y + z$ yrs.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles x + y + z
Exercise 10, problem 9
What is the sum of $m$ taken $b$ times?
Printed answer:
- $bm$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles b*m

Exercise 12

Exercise 12, problem 1
Express the double of $x$.
Printed answer:
- $2x$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise 12, problem 10
Express the product of $a$ factors each equal to $d$.
Printed answer:
- $d^a$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise 12, problem 11
Write the second power of $a$ added to three times the cube of $m$.
Printed answer:
- $3m^3 + a^2$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise 12, problem 12
Express $x$ to the power $2m$, plus $x$ to the power $m$.
Printed answer:
- $x^{2m} + x^m$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise 12, problem 13
What is the interest on $x$ dollars for $m$ years at 6 %?
Printed answer:
- $\frac{6}{100}mx$ dols., or $6mx$ cts.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation 6*m*x/100
Exercise 12, problem 14a
In a certain school there are $c$ girls, and three times as many boys less 8. How many boys, and how many boys and girls together?
Printed answer:
- $3c - 8$ boys; $4c - 8$ boys and girls.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation 3*c - 8
Exercise 12, problem 14b
In a certain school there are $c$ girls, and three times as many boys less 8. How many boys, and how many boys and girls together?
Printed answer:
- $3c - 8$ boys; $4c - 8$ boys and girls.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation 4*c - 8
Exercise 12, problem 15
If $x$ men can do a piece of work in 9 days, how many days would it take 1 man to perform the same work?
Printed answer:
- $9x$ days.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation 9*x
Exercise 12, problem 16
How many thirds are there in $x$?
Printed answer:
- $3x$ thirds.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation 3*x
Exercise 12, problem 17
How many fifths are there in $b$?
Printed answer:
- $5b$ fifths.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation 5*b
Exercise 12, problem 18
A man bought a horse for $x$ dollars, paid 2 dollars a week for his keeping, and received 4 dollars a week for his work. At the expiration of $a$ weeks he sold him for $m$ dollars. How much did he gain?
Printed answer:
- $m - x + 2a$ dols.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation m - x + 2*a
Exercise 12, problem 19
James has $a$ walnuts, John twice as many less 8, and Joseph three times as many as James and John less 7. How many have all together?
Printed answer:
- $12a - 39$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation 12*a - 39
Exercise 12, problem 2
Express the product of $x, y$, and $z$.
Printed answer:
- $xyz$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise 12, problem 3
How many cents in $x$ dollars?
Printed answer:
- $100x$ cts.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise 12, problem 4
Write $a$ times $b$ times $c$.
Printed answer:
- $abc$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise 12, problem 5
What will $a$ quarts of cherries cost at $d$ cents a quart?
Printed answer:
- $ad$ cts.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise 12, problem 6
If a stage coach, goes $b$ miles an hour, how far will it go in $m$ hours?
Printed answer:
- $mb$ miles.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise 12, problem 7
In a cornfield there are $x$ rows, and $a$ hills in a row. How many hills in the field?
Printed answer:
- $ax$ hills.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise 12, problem 8
Write the cube of $x$.
Printed answer:
- $x^3$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise 12, problem 9
Express in a different way $a \times a \times a \times a \times a \times a \times a \times a \times a$.
Printed answer:
- $a^9$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**9

Exercise 13

Exercise 13, problem 1
Express five times $a$ divided by three times $c$.
Printed answer:
- $\frac{5a}{3c}$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 5*a/(3*c)
Exercise 13, problem 10
A boy who earns $b$ dollars a day spends $x$ dollars a week. How much has he at the end of 3 weeks?
Printed answer:
- $18b - 3x$ dols.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 18*b - 3*x
Exercise 13, problem 11a
A can perform a piece of work in $x$ days, B in $y$ days, and C in $z$ days. Express the part of the work that each can do in one day. Express what part they can all do in one day.
Printed answer:
- $\displaystyle \frac{1}{x}$
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 1/x
Exercise 13, problem 11b
A can perform a piece of work in $x$ days, B in $y$ days, and C in $z$ days. Express the part of the work that each can do in one day. Express what part they can all do in one day.
Printed answer:
- $\displaystyle \frac{1}{y}$
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 1/y
Exercise 13, problem 11c
A can perform a piece of work in $x$ days, B in $y$ days, and C in $z$ days. Express the part of the work that each can do in one day. Express what part they can all do in one day.
Printed answer:
- $\displaystyle \frac{1}{z}$
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 1/z
Exercise 13, problem 11d
A can perform a piece of work in $x$ days, B in $y$ days, and C in $z$ days. Express the part of the work that each can do in one day. Express what part they can all do in one day.
Printed answer:
- $\displaystyle \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 1/x + 1/y + 1/z
Exercise 13, problem 12
How many square feet in a garden $a$ feet on each side?
Printed answer:
- $a^2$ sq. ft.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles a**2
Exercise 13, problem 13
A money drawer contains $a$ dollars, $b$ dimes, and $c$ quarters. Express the whole amount in cents.
Printed answer:
- $100a + 10b + 25c$ cts.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 100*a + 10*b + 25*c
Exercise 13, problem 14
$x$ is how many times $y$?
Printed answer:
- $\displaystyle \frac{x}{y}$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles x/y
Exercise 13, problem 15
If $m$ apples are worth $n$ chestnuts, how many chestnuts is one apple worth?
Printed answer:
- $\displaystyle \frac{n}{m}$ chestnuts.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles n/m
Exercise 13, problem 16
Divide 30 apples between two boys so that the younger may have two-thirds as many as the elder.
Printed answer:
- 12; 18 apples.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {y: 12, x: 18}
Exercise 13, problem 2
How many dollars in $y$ cents?
Printed answer:
- $\frac{y}{100}$ dols.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles y/100
Exercise 13, problem 3
How many books at $a$ dimes each can be bought for $x$ dimes?
Printed answer:
- $\frac{x}{a}$ books.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles x/a
Exercise 13, problem 4
How many days will a man be required to work for $m$ dollars if he receive $y$ dollars a day?
Printed answer:
- $\frac{m}{y}$ days.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles m/y
Exercise 13, problem 5
$x$ dollars were given for $b$ barrels of flour. What was the cost per barrel?
Printed answer:
- $\frac{x}{b}$ dols.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles x/b
Exercise 13, problem 6
Express $a$ plus $b$, divided by $c$.
Printed answer:
- $\frac{a + b}{c}$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles (a + b)/c
Exercise 13, problem 7
Express $a$, plus $b$ divided by $c$.
Printed answer:
- $a + \frac{b}{c}$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles a + b/c
Exercise 13, problem 8
A man had $a$ sons and half as many daughters. How many children had he?
Printed answer:
- $a + \frac{a}{2}$, or $\frac{3}{2}a$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles a + a/2
Exercise 13, problem 9
If the number of minutes in an hour be represented by $x$, what will express the number of seconds in 5 hours?
Printed answer:
- $300x$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 300*x