Exercise 8

1. A man bought 7 head of cattle at $32.50 a head and sold them at $54 a head. How much did he make on them?

2. A certain number of articles are bought at a certain price each and sold at a certain higher price each. In what two ways can you find the total profit? Which is the easier?

3. If you are told how much a boy earns each week and how much he spends each week, how can you find out how much he saves in a year?

4. If you are told how much a man earns in a year, what else do you need to know in order to find out how much he earns each day he works? If you are told this, how do you then find out how much he earns a day? If told how many hours a day he works, how can you then find out how much he earns an hour? In a minute?

5. If told the number of oranges in a box and the selling price a dozen, how can you find the selling price of the box of oranges?

6. What do you need to know in order to find out how many steps you take in walking a mile?

7. How can you find out how many steps you take in walking a mile if you know how many feet you go at each step?

8. What measurement would you make to find how many revolutions a wheel makes in going a mile?

9. The circumference of a certain wheel is 12 feet. How many revolutions does it make in going a hundred yards?

10. If told that two-thirds of the pupils in your room are girls, what else would you need to know in order to compute the number of boys in the room?

11. If you know the cost of several tons of coal and the cost of one ton, how can you find the number of tons bought?

12. If told the width of a piece of cloth, how can you find the number of strips, each 4 inches wide and as long as the piece, which can be cut from it?

13. How can you find out how many cords in a pile of four-foot wood?

14. What measurements would you make to find how many books all of the same size a certain shelf would hold? 15. How can you find the average of 4 numbers?

11. The cost formula.

1. If a yard of cloth costs 83, how much do 9 yards cost? 2. If the price of hay is $12.50 a ton, how much do 2 tons cost?

3. If the price of an article is known, how is the cost of a certain number of these articles found?

4. If p represents the price of an article and n is the number bought, what is the cost of the n articles?

The cost of a number of articles is equal to the number bought times the price of each.

If the cost is represented by C then this principle may be condensed into the formula:

C=nXp.

This is a short form for the statement, cost equals the number times the price of one.

Such a short form is called a formula.

Formulas are especially useful in solving a large number of problems of the same kind.

EXAMPLE. Find the cost, C, if n=3 and p=5¢.

SOLUTION. C=nXp=3×5¢ = 15¢.

When numbers are represented by letters their product is usually expressed by writing them together with no sign between them. Thus, np means n times p; 5x means 5 times x.

Exercise 9

Use the formula C=np to find the values of C for the given

9. A woman buys three cantaloupes for a quarter. What is the average price?

10. Fifteen tons of coal cost $48. What is the price per ton?

11. If told the cost of a certain number of things, how may the price of one be found? Make a formula of the rule you have just stated.

This formula may be written in either of two ways:

p=C÷n, read "p equals C divided by n";

or p=C, read "p equals C over n," or "p equals C divided

by n."

Complete this table, giving values of p for the given values

17. Read and write the product of x and y; of 7 and c;

of m, n, and k; of r, s, and 3; of t, 9, and d.

12. Making formulas. Since it is generally much easier to remember a formula than the much longer principle or rule, most of the rules of arithmetic are stated as formulas.

The rule for finding the cost of a number of articles when the price of one and the number of articles are known has been stated in the formula C = np.

To state a rule as a formula it is necessary to represent the numbers mentioned in the rule by letters, and to indicate the processes to be performed by the proper signs.

The sum of a and b is written a+b. The difference when b is subtracted from a is written a-b. The product of a and b is written either a×b or ab, generally the latter. The quotient of a by b is written either a÷b or

a

b

EXAMPLE. One-fourth of the desks were taken from a school building having five rooms. The first room contained n desks, the second 15 desks, and each of the other three rooms contained d desks. How many desks were taken away?

SOLUTION.

In the last three rooms there were 3d desks. In all the five rooms there were 3d+n+15 desks.

1. Write the sum of c and d ; of m, n, and r; of 2 and m;

of 3, a, and b; of 7, b, c, and x.

Read each of these sums.

2. In one field there are 12 acres and in another n acres. How many acres in both fields?

3. There are n children in one grade. Seven of them are boys. How many girls are there?

4. Write the number which is twice k.

5. How many inches in 6 ft.? In fft.?

6. How many ears have h horses? How many feet?

7. How many feet have m men? How many feet have h horses and m men?

8. How many quarts in n gallons? How many gallons in n quarts?

9. How many feet in y yd. 1 ft.? How many inches?

10. How many oranges can be bought for 80¢ at r cents apiece? For r cents at 5 cents apiece?

11. If the distance around a square is p, how long is one side?

In making formulas, first think the method by which the number required for the answer may be found; then, using a letter to represent the number required, indicate the processes by which it is found.

EXAMPLE 1.

Make a formula for finding the number of inches in ƒ ft. n in.

SOLUTION. Let i represent the number of inches in ƒ ft. n in. To find this number of inches we must multiply the number of feet by 12 and add the number of inches.

EXAMPLE 2. Make a formula for finding the total weight of a basket containing 3 buckets of lard and 5 cans of corn, if the weight of the empty basket, the weight of one can of lard, and the weight of one can of corn are known.

SOLUTION. To find the total weight we must multiply the weight of one bucket of lard by 3, the weight of one can of corn by 5, and add these two weights to the weight of the basket. Represent the weight of the empty basket by b, the weight of one bucket of lard by l, the weight of one can of corn by c, and the total weight by W. Then the weight of the 5 cans of corn is 5 c, the weight of the 3 buckets of lard is 3l, and the weight of the empty basket is b. The total weight, W, is therefore the sum of b, 31, and 5 c.

The formula is, therefore, W=b+31+5c.