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1. Name a number that will exactly contain 6 and 9; 8 and 12; 7 and 9.

A common multiple of two or more numbers is a number that is exactly divisible by each of them; thus, 36 is a common multiple of 6 and 9.

2. Name the least number that is exactly divisible by 6 and 9; by 8 and 12.

The least common multiple (1. c. m.) of two or more numbers is the least number that is exactly divisible by each of them; thus, 18 is the 1. c. m. of 6 and 9.

3. Name the 1. c. m. of 6 and 8; of 9 and 12; of 8 and 12.

Written Work

1. Find the 1. c. m. of 18, 32, and 40.

18=2x3x3.

32=2x2x2x2x2

40=2x2x2x5

1. c. m. 25 x 32 x 5, or 1440.

=

The 1. c. m. of two or more numbers is the product of all their prime factors, each factor being used as often as it occurs in any number.

2 occurs 5 times as a factor in 32. It must, therefore, be used 5 times in the 1. c. m. 3 occurs twice as a factor in 18; it must, therefore, be used twice in the 1. c. m. 5 occurs once as a factor in 40; it must, therefore, be used once in the 1. c. m. Hence the 1. c. m. of 18, 32, and 40 is 25 × 32 × 5 = 1440.

2. Find the 1. c. m. of 12, 36, 54, and 63. 2)12 36 54 63

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1. c. m. = 22 x 33 x7 = 756.

Since 12 is a divisor of 36, the l. c. m. of 36, 54, and 63 is also a multiple of 12. 12 may therefore be rejected from the work.

Divide any two of the numbers by Then divide the quotients in like manner until The product of the divisors and

a common prime factor.
the quotients are prime to each other.
the last quotients is the 1. c. m,

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144 36 4. =

CANCELLATION

We may separate the dividend 144 into the factors 9 and 16, and the divisor 36 into the factors 9 and 4. We may, therefore, write 144 ÷ 36 = 4 as follows :

(9 × 16) ÷ (9 × 4) = 4.

By striking out the common factor 9 in both dividend and divisor, the problem is: (16 ÷ 4) = 4.

Striking out equal factors from both dividend and divisor does not change the quotient.

When the product of a number of factors is to be divided by the product of another set of factors, the usual way is to write the dividend above a line and the divisor below, and strike out equal factors: :

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Cancellation is the process of shortening operations in division by striking out equal factors from both dividend and divisor.

Written Work

1. Divide 3 × 6 × 8 × 20 by 11 × 4 × 20.

2

3 × 6 × 8 × 20_36

=

11 × 4 × 20 11

3

=

Write the dividend above and the divisor below a line. First cancel the 20 from dividend and divisor. Then cancel the

factor 4 from 8 in the dividend and from 4 in the divisor, leaving 2 in the dividend and 1 in the divisor. As there are no other factors common to dividend and divisor, you have 3 × 6 × 2, or 36, divided by 11, or if, which equals 31oí·

NOTE. When equal factors in the terms are canceled, the factor 1 always remains, but as it does not affect the product, it need not be written.

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7. 148 × 64 × 57 × 12 by 114 × 32 × 48

8. By selling butter at 24 per pound a lady receives enough money to buy 48 pounds of coffee at 20 per pound. How many pounds of butter does she sell?

9. A man worked 16 days of 10 hours each at 20¢ per hour, and spent the money he received for corn at 40 per bushel. How many bushels of corn did he get?

FRACTIONS

FRACTIONAL UNITS

When we say 8, 6 ft., $2, 6 rd., 7 mi., 5 in., what are the units of measure?

Observe that in each case the number and its unit of measure are of the same denomination; thus, 1 ft. is the unit of measure in 6 ft.

A unit, therefore, is any single quantity with which another quantity of the same kind is measured or compared; as, 1 is the unit of 10; 1 ft. is the unit of 8 ft.; 1 yd. of 2 yd.; 1 mi. of 12 mi.; 1 acre of 5 acres, etc.

A fractional unit is one of the equal parts into which an integral unit has been divided; as, o,,,, 15, etc. A fraction is one or more fractional units; as, 1, 1, 7, 1, , etc.

The terms of a fraction are the numerator and the denominator.

The denominator indicates the size of the fractional unit; it is written below the line, and shows into how many parts the integral unit has been divided. Thus, in the fraction, 5 is the denominator, and shows that some unit has been divided into 5 parts.

The numerator indicates the number of fractional units; it is written above the line, and shows how many parts are taken. Thus, in the fraction , 4 is the numerator, and

shows that 4 parts have been taken.

1. What is the fractional unit in ? ? ? ?

2. Read the following fractional units in order of their size, beginning with the largest: 16, 1, 4, 12, 1, 4, and 21.

3. The use of the numerator and the denominator in 12 yd. may be explained thus, 2 yd. = 9 × 12 yd.

As the integral unit is the basis by which we measure whole numbers, so the fractional unit is the basis by which we measure fractions of the same kind.

4. Name the unit of 4 ft.; 5 mi.; ;

.

5. Which is the larger, or 1? or 1? or 1? Explain how much larger in each case.

6. What is the difference in value between the fraction g and an integral unit?

A common fraction is a fraction that has both terms expressed; as, 4, 4, 1.

A proper fraction is a fraction less in value than 1; as, 1, 3, 1, 1, 8, 4, 4, etc.

3 4

An improper fraction is a fraction equal to or greater in value than 1; as, &, f, 1, §, &, 0, etc.

9 4 5

A mixed number is a number expressed by a whole number and a fraction; as, 31⁄2, 123.

Change each of the following to integral units, or to mixed numbers.

Thus, = 1.

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