II.

Now, we know what James had. He had oranges, and we are to discover simply the number of oranges he had.

2. James and John together have $$24$, and James has $$8$ more than John. How many dollars has each?

lint Let $x$ stand for the number of dollars John has; lintthen $x+8$ is the number of dollars James has; lintand $x+(x+8)$ is the number of dollars they both have.

But $24$ is the number of dollars they both have. $\therefore x+(x+8)=24.$

Removing the parenthesis, $x+x+8=24.$

2x = 16. lint Dividing by $2$, x = 8. lint Since $x=8$, x + 8 = 16.

Therefore John has $$8$, and James has $$16$.

The beginner must avoid the mistake of writing $\text{{}Let$x = {}$Joh{n}^{\prime }smoney.\}$

We are required to find the number of dollars John has, and therefore $x$ must represent this required number.

3. The sum of two numbers is $18$, and three times the greater number exceeds four times the less by $5$. Find the numbers.

Let $x=$the greater number.

Then, since $18$ is the sum and $x$ is one of the numbers, the other number must be the sum minus $x$. Hence $18-x=\text{the smaller number}.$

Now, three times the greater number is $3x$, and four times the less number is $4(18-x)$.

lint Hence, 3x - 4(18 - x) = the excess. lint But 5 = the excess, 3x - 4(18 - x) = 5 3x - (72 - 4x) = 5, lintor 3x - 72 + 4x = 5. 7x = 77, lintand 0 x = 11.

Therefore the numbers are $11$ and $7$.

Exercise 10.

1. If a number is multiplied by $9$, the product is $270$. Find the number.

2. If the sum of the ages of a father and son is $60$ years, and the father is $5$ times as old as the son, what is the age of each?

3. The sum of two numbers is $91$, and the greater is $6$ times the less. Find the numbers.

4. A tree $90$ feet high was broken so that the part broken off was $8$ times the length of the part left standing. Find the length of each part.

5. The difference of two numbers is $7$, and their sum is $53$. Find the numbers.

6. The difference of two numbers is $12$, and their sum is $84$. Find the numbers.