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20. There are three numbers such that 1 taken from the first will give the second, the second multiplied by 3 will give the third, and their sum is equal to 26. What are the numbers?

21. John and Charles together have just 31 oranges. If 1 be taken from John's, and the remainder be multiplied by 5, the product will be equal to Charles's number. How many has each?

22. A basket is filled with apples, lemons, and oranges; in all, 26. The number of lemons exceed the number of apples by 2, and the number of oranges is double that of the lemons. How many are there of each?

LESSON XI.

1. John has a certain number of apples, the half of which is equal to 10. How many has he?

Let x denote the number of apples. Then x divided by 2 is equal to 10. If one half of x is equal to 10, twice one half of x, or x, is equal to twice 10, which is 20: hence x is equal to 20.

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NOTE. A similar analysis is applicable to any one of the fractional units. Let each question be solved according to the analysis.

2. John has a certain number of oranges, and one third of his number is 15. How many has he?

3. If one fifth of a number is 6, what is the number?

4. If one twelfth of a number is 9, what is the number? 5. What number added to one half of itself will give a sum equal to 12?

Denote the number by x. Then x plus one half of x equals 12. But x plus one half of x equals three halves of x: hence three halves of equal 12. If three halves of x equal 12, one half of x equals one third of 12, or 4. If one half of x equals 4, x equals twice 4, or 8: therefore x equals 8. Hence we write,

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6. What number added to one third of itself will give a sum equal to 12?

7. What number added to one fourth of itself will give a sum equal to 20?

8. What number added to a fifth of itself will make 24 ?

9. What number diminished by one half of itself will leave 4? Why?

10. What number diminished by one third of itself will leave 6?

11. James gave one seventh of his marbles to William, and then had 24 left. How many had he at first?

12. What number added to two thirds of itself will give a sum equal to 20?

13. What number diminished by three fourths of itself will leave 9?

14. What number added to five sevenths of itself will make 24?

15. What number diminished by seven eighths of itself will leave 4?

16. What number added to eight ninths of itself will make 34?

17. What number diminished by eleven twelfths of itself will leave 5?

18. Margaret gave nine tenths of her apples to her sister, and then had 6 left. How many had she at first?

19. What number added to 3 times one ninth of itself will give 72?

20. Henry had a certain number of cents. He lost one third of them, and had 15 left. How many had he at first?

CHAPTER I.

DEFINITIONS.

1. Quantity is anything which can be increased, diminished, and measured; as number, distance, weight, time, etc. To measure a thing is to find how many times it contains some other thing of the same kind, taken as a standard. assumed standard is called the unit of measure.

The

2. Mathematics is the science which treats of the measurement, properties, and relations of quantities.

In pure mathematics there are but eight kinds of quantity, and consequently but eight kinds of units; viz., units of number, units of currency, units of length, units of surface, units of volume, units of weight, units of time, and units of angular measure.

3. Algebra is a branch of mathematics in which the quantities considered are represented by letters, and the operations to be performed are indicated by signs.

4. The quantities employed in algebra are of two kinds,known and unknown.

Known quantities are those whose values are given. They are generally represented by the leading letters of the alphabet; as, a, b, c, etc.

Unknown quantities are those whose values are required. They are generally represented by the final letters of the alphabet; as, x, y, z, etc.

When an unknown quantity becomes known, it is often denoted by the same letter, with one or more accents; as,

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x', x", "". These symbols are read, "x prime,' x third."

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x second,"

5. The sign of addition (+) is called plus. When placed between two quantities, it indicates that the second is to be added to the first. Thus, a+b is read "a plus b," and indicates that b is to be added to a. If no sign is written, the signis understood.

The sign is sometimes called the positive sign; and the quantities before which it is written are called positive quantities, or additive quantities.

6. The sign of subtraction (-) is called minus. When placed between two quantities, it indicates that the second is to be subtracted from the first. Thus, the expression c- - d, read "c minus d," indicates that d is to be subtracted from c. If a stands for 6, and d for 4, then ad is equal to 6-4, which is equal to 2.

The sign is sometimes called the negative sign; and the quantities before which it is written are called negative quantities, or subtractive quantities.

7. The sign of multiplication (×) is read "multiplied by," or "multiplied into." When placed between two quantities, it indicates that the first is to be multiplied by the second. Thus, ab indicates that a is to be multiplied by b. If a stands for 7, and b for 5, then a x b is equal to 7 x 5, which is equal to 35.

The multiplication of quantities is also indicated by simply writing the letters one after the other, and sometimes by placing a point between them. Thus, a b signifies the same thing as ab or as a.b; a xbx c signifies the same thing as abc or as a.b.c.

8. A factor is any one of the multipliers of a product. Factors are of two kinds, numeral and literal. Thus, in the

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