Lesson 2 · from §3 of the 1910 text
The Usefulness of Symbols
Symbols let you abbreviate ordinary language when solving problems.
For example: Three times a certain number is equal to 20 diminished by 5. What is the number?
If n represents the number, the statement and question can be written in symbols:
3n = 20 - 5, \qquad n = \,?
The symbolic statement 3n = 20 - 5 is called an equation, and n is the unknown number. Solving it:
\begin{aligned} 3n &= 20 - 5 \\ 3n &= 15 \\ n &= 5 \end{aligned}
Quick check
Solve the same way. If 4n = 30 - 6, what is n?
This is the algebraic method of stating and solving a problem. It's brief and direct, and its advantages will become clearer as you go.
Worked example
The sum of two numbers is 112. The greater is three times the less. What are the numbers?
By the conditions of the problem, greater number + less number = 112. If we represent the less by l, then 3l must represent the greater, and the statement becomes
\begin{aligned} 3l + l &= 112 \\ 4l &= 112 && \text{(collecting)} \\ l &= \tfrac{112}{4} = 28 \\ 3l &= 3 \times 28 = 84 \end{aligned}
So the greater number is 84 and the less is 28.
Worked example: interest
What sum of money placed at interest for 1 year at 5% amounts to $378?
From arithmetic, principal + interest = amount. If p represents the principal,
\begin{aligned} p + 0.05p &= 378 \\ 1.05p &= 378 && \text{(collecting)} \\ p &= \tfrac{378}{1.05} = 360 \end{aligned}
So the required sum is $360.
Quick check
What sum of money placed at interest for 1 year at 4% amounts to $520?
Worked example: rate of interest
At what percent simple interest will $375 gain $75 in 2 years?
Let x be the rate of interest. Then 375x is the interest for one year and 750x is the interest for two years. But the interest for two years is $75, so
750x = 75, \qquad x = \tfrac{1}{10}.
So the money is lent at 10%.
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
The sum of two numbers is 160. The greater is four times the less. Find each.
A certain number plus five times itself equals 216. Find the number.
One number is seven times another. Their sum is 72. Find each.
The first of three numbers is twice the third, and the second is four times the third. The sum of the three numbers is 105. Find the numbers.
The sum of three numbers is 117. The second is twice the first, and the third three times the second. Find each.
There are three numbers whose sum is 192. The first is twice the second, and the third equals the sum of the other two. Find the numbers.
The sum of three numbers is 324. The second is five times the first, and the third is six times the second. Find the numbers.
The sum of three numbers is 104. The second is three times the first, and the third is the sum of the other two. Find the numbers.
What sum of money placed at interest for 1 year at 6% amounts to $265?
What sum of money placed at simple interest for 3 years at 4% will amount to $700?
In how many years will $225, at 6% simple interest, gain $27?
In how many years will $520, at 6\tfrac{1}{2}% simple interest, gain $169?
At what percent simple interest will $825 gain $165 in 4 years?
At what percent simple interest will $250 amount to $317.50 in 6 years?
In how many years will $200 double itself at 4% simple interest?
In how many years will $150 treble itself at 5% simple interest?
The perimeter of a certain square is 160 feet. Find the length of each side.
The perimeter of a certain rectangle is 256 feet. It is three times as long as it is wide. Find its dimensions.
The perimeter of the rectangle formed by placing two equal squares side by side is 198 inches. Find the dimensions and the perimeter of each square.
Two equal squares are placed side by side, forming a rectangle. If the perimeter of each square is 120 inches, find the perimeter of the rectangle.