Part 218

Now I will write these--

1 1 - + - = 2 2

1 ÷ 2 =

1 1 less - = 2

1 1 - - = 2

1 2 × - = 2 1 1 divided by - = 2

Give me the answers and I will write them.

Drawings showing the “placing” of disks for number combinations can then be made; as,

Make similar drawings to tell about halves.

Proceed like this--How many halves in a pie? If a pie cost 10 cents, what will half a pie cost? Who can tell other stories about halves? etc.

Learn fourths along with halves.

LEARNING THE FRACTION 1/3 AND OTHERS WITH DISKS

Cut several disks into thirds and have children practice on cutting, so that they will be able to make the three parts of each disk equal. Frequently children will find pleasure in “teaching” one another.

Then proceed like this: What do you call each of these parts? Why are they called thirds? How many thirds in a circle? I am going to take a circle and cut it any way, so as to make three parts; do I call these unequal parts thirds? Why not? Let me write one-third on a piece of paper for you. (Write, 1/3.) Draw a circle for me. Instead of cutting it, draw lines where you would cut it to make thirds. Write one-third (1/3) on each third of a circle. I write this (1/3 + 1/3). Who can tell me what the answer is? Are two-thirds and two-thirds more than one? How much more? I have two-thirds of an apple and give Mary one-third, how much have I left? Who can give other story problems about thirds? Everybody try, etc.

Learn sixths along with thirds. Use disks, dots, marks, sticks, and inches to illustrate.

Remember that no advance should be made until each little part is understood. Then have fifths compared with fourths, thirds, and halves.

Teach tenths along with fifths.

When twelfths are taught, show the relations between twelfths and sixths, fourths, thirds, and halves.

EQUAL FRACTIONS IN DIFFERENT FORMS

Have the children see how fractions may differ in form but still remain the same in value.

Begin with his knowledge of smaller fractions as

1 2 3 4 5 -, -, -, -, and -- of an apple. 2 4 6 8 10

Let them show by the use of drawings that fractions may have large or small terms but be equal in value.

Write a number of proper fractions, improper fractions, and mixed numbers, and have the children pick out those of each kind; as,

3 5 19 20 18 11 -, 27-1/2, --, --, --, --, --, 3-1/2, 16-2/3 8 11 20 20 16 5

PRINCIPLES OF FRACTIONS

_1. A fraction’s value is the quotient obtained by dividing the numerator by the denominator._

6 6 - = 3 3 is the value of - 2 2

2 2 2 2 - = - - is the value of - 3 3 3 3

_2. Multiplying the denominator of a fraction divides the fraction by that number._

1 1 - = - 2 × 4 8

3 3 - = -- 7 × 3 21

2 2 - = -- 3 × 9 27

_3. Dividing the denominator of a fraction multiplies the fraction by that number._

3 3 - = - 8 ÷ 4 2

10 10 -- = -- 9 ÷ 3 3

3 3 -- = - 10 ÷ 5 2

_4. Multiplying the numerator of a fraction multiplies the fraction by that number._

2 × 2 4 - = - 3 3

1 × 8 8 - = - 9 9

5 × 3 15 - = -- 8 8

_5. Dividing the numerator of a fraction divides the fraction by that number._

4 ÷ 2 2 - = - 7 7

12 ÷ 12 1 -- = -- 16 16

3 ÷ 3 1 - = - 7 7

_6. Multiplying both numerator and denominator of a fraction by the same number does not change the value of the fraction._

1 × 3 3 1 - = - = - 3 × 3 9 3

6 × 2 12 6 - = -- = - 7 × 2 14 7

_7. Dividing both numerator and denominator of a fraction by the same number does not change the value of the fraction._

12 ÷ 3 4 12 -- = - = -- 15 ÷ 3 5 15

18 ÷ 9 2 18 -- = - = -- 27 ÷ 9 3 27

REDUCTION OF FRACTIONS

is the process of changing their forms without altering their values.

To reduce a fraction to its lowest terms:

RULE.--Divide both terms by their greatest common divisor.

Reduce 8/12 to its lowest terms.

WORK: 4) 8/12 (2/3 _Ans._ 2/3

Four is the G. C. D. of 8 and 12; hence 8/12 ÷ 4 = 2/3.

Reduce 35/56 to its lowest terms.

WORK: 7) 35/56 (5/8 _Ans._ 5/8

Seven is the G. C. D. of 35 and 56; hence 35/56 ÷ 7 = 5/8.

A fraction whose terms have no common divisor is in its lowest terms, as 9/16.

To reduce an improper fraction to a whole or mixed number:

RULE.--Divide the numerator by the denominator; the quotient will be the whole or mixed number.

How many units in 30/6?

WORK: 30 ÷ 6 = 5 _Ans._ 5.

There are as many units in 30 sixths as 6 is contained times in 30.

Reduce 75/4 to a mixed number.

WORK: 75 ÷ 4 = 18 + 3 _Ans._ 18-3/4.

In 75 fourths there are 18 units, and 3 fourths over, which equals 18-3/4.

To reduce a mixed number to an improper fraction:

RULE.--Multiply the whole number by the denominator of the fraction; add the numerator to the product, and write the sum over the denominator.

Reduce 18-3/4 to an improper fraction.

WORK: 18 × 4 = 72

72/4 + 3/4 = 75/4

_Ans._ 75/4.

In 18 are 72 fourths, plus the 3 fourths, equals 75 fourths.

To reduce two or more fractions to their least common denominator:

RULE.--Find the least common multiple of the given denominators for a common denominator. Then for each new numerator take such a part of this common denominator as the fraction is part of 1.

Reduce 1/2, 2/3 and 3/4 to their L. C. D.

WORK:

1 6 - = -- 2 12

2 8 - = -- 3 12

3 9 - = -- 4 12

_Ans._ 6/12, 8/12 and 9/12.

The L. C. M. of the denominators 2, 3 and 4 is 12. Hence, 12 is the L. C. D. to which the given fractions can be reduced. Then to change 1/2 to 12ths, say, 1/2 of 12 is 6, and write it over 12; to change 2/3 to 12ths, say 2/3 of 12 is 8, and write it over 12; to change 3/4 to 12ths, say, 3/4 of 12 is 9, and write it over 12.

Fractions must be reduced to a common denominator to be added or subtracted.

ADDITION OF FRACTIONS

If two or more fractions have the same denominator, their sum is obtained by adding the numerators.

WORK: 1 4 5 1 + 4 + 5 10 3 - + - + - = --------- = -- = 1- 7 7 7 7 7 7

If the fractions have different denominators, we must first express them as equivalent fractions with the same denominator.

EXAMPLE 1: Find the value of

1 3 5 2 - + - + -- + - 9 7 21 3

The lowest common multiple is 63. The several denominators, when divided into 63, give 7, 9, 3, 21 respectively, for quotients. Therefore, we multiply the numerators and denominators of the fractions by 7, 9, 3, 21, and add the numerators to obtain the required sum. The result must be reduced to a mixed number or to lower terms, if necessary.

WORK:

1 3 5 2 - + - + -- + - 9 7 21 3

7 + 27 + 15 + 42 = ---------------- 63

= 91/63 = 1-28/63 = 1-4/9 _Ans._

In adding mixed numbers, first add the whole numbers, then the fractions, finally adding the two results.

EXAMPLE 2: Add together 3-1/8 + 7/24 + 7-11/15 + 4-3/20. Given expression:

1 7 11 3 = 3 + 7 + 4 + - + -- + -- + -- 8 24 15 20

15 + 35 + 88 + 18 = 14 + ----------------- 120

156 36 3 = 14 + --- = 14 + 1--- = 15-- _Ans._ 120 120 10

SUBTRACTION OF FRACTIONS

The principle is the same as in addition. Reduce the fractions, if they have different denominators, to a common denominator, and then take the difference of the numerators. In the case of mixed numbers, subtract the whole numbers and the fractions separately.

EXAMPLE 1: Take 4-5/21 from 6-3/7.

3 5 3 5 6- - 4-- = 6 - 4 + - - -- 7 21 7 21

9 - 5 = 2 + ----- 21

4 4 = 2 + -- = 2-- _Ans._ 21 21

If the fractional part of the number to be subtracted be greater than the fractional part of the other number, we proceed as follows:

EXAMPLE 2: From 7-4/15 take 4-11/25.

4 11 4 11 7-- - 4-- = 7 - 4 + -- - -- 15 25 15 25

20 - 33 = 3 + ------- 75

75 + 20 - 33 = 2 + ------------ 75

62 62 = 2 + -- = 2-- _Ans._ 75 75

EXAMPLE 3: Simplify 3-2/9 + 4-5/7 - 5-13/21 + 2/35 - 1-14/15. Given expression:

2 5 13 2 14 = 3 + 4 - 5 - 1 + - + - - -- + -- - -- 9 7 21 35 15

70 + 225 - 195 + 18 - 294 = 1 + ------------------------- 315

313 - 489[15] = 1 + --------- 315

628 - 489 139 = --------- = --- _Ans._ 315 315

[15] Obtained by adding all the numerators with + before them, and then all those with - before them.

MULTIPLICATION OF FRACTIONS

(i) When the multiplier is a whole number. This, as in the case of whole numbers, means that we have to find the sum of a given number of repetitions of the fraction.

EXAMPLE 1:

7 7 7 7 7 28 - × 4 means - + - + - + -, _i.e._, --; or 9 9 9 9 9 9

7 × 4 ----- 9

Hence, to multiply a fraction by a whole number, simply multiply the numerator by that number.

Since the multiplier thus becomes a factor of the numerator, we cancel any common factors contained in the multiplier and the denominator; and this may be done before we perform the actual multiplication:

EXAMPLE 2: Multiply 19/46 by 69.

19 19 × 69 19 × 3 57 -- × 69 = ------- + ------ (cancelling 23), = -- = 28-1/2 _Ans._ 46 46 2 2

It follows that if the multiplier be itself a factor of the denominator, we may, to multiply a fraction by a whole number, divide the denominator by that number.

(ii) When the multiplier is a fraction.

EXAMPLE: In performing the operation 7 × 9, it is plain that we do to 7 what we do to a unit to obtain 9. Similarly, 3/5 × 4/11 may be looked upon as doing to 3/5 what we do to the unit to obtain 4/11.

Now, to obtain 4/11 from the unit, we must divide the unit into 11 equal parts and take 4 of them.

Therefore, to find the value of 3/5 × 4/11 we must divide 3/5 into 11 equal parts and take 4 of them.

But 3/5 = 33/55 = 3/55 × 11, so that, the eleventh part of 3/5 is 3/55; and, if we take 4 of these parts, we get 3/55 × 4 or 12/55.

3 4 12 Thus, - × -- = --. Now 12 = 3 × 4, and 55 = 5 × 11. 5 11 55

Hence we have the following rule: To multiply two fractions together, multiply the numerators for a new numerator and the denominators for a new denominator.

As in Example 2 the work is shortened if we cancel common factors from the numerators and denominators.

EXAMPLE: Multiply 22/91 by 13/77.

2 2̸2̸ × 1̸3̸ 2 The product = -------- = -- _Ans._ 9̸1̸ × 7̸7̸ 49 7 7

Here, the 22 of the numerator and the 77 of the denominator contain a common factor, 11. Therefore, we cross out the 22 and write 2 above it, and cross out the 77 and write 7 under it. Similarly, we cancel the factor 13 from 13 and 91. There is now 2 left for numerator and 7 × 7 for denominator.

To multiply more than two fractions together, we proceed in the same way.

In multiplication of fractions, mixed numbers must first be expressed as improper fractions.

EXAMPLE: Simplify 5-1/7 × 11/27 × 1-11/24.

3̸ 5 3̸6̸ 11 3̸5̸ 55 1 Given expression = -- × -- × -- = -- = 3-- _Ans._ 7̸ 2̸7̸ 2̸4̸ 18 18 9 2

DIVISION OF FRACTIONS

(i) When the divisor is a whole number. Suppose we have to divide 7/9 by 4.

We know 7/9 = 28/36. This fraction means that the unit is divided into 36 equal parts, and 28 of the parts taken. If we divide the 28 parts by 4, we get 7 of them--_i.e._ 7/36. Hence 7/9 ÷ 4 = 7/36.

Therefore, to divide a fraction by a whole number, we multiply the denominator by that number.

In the same way as already explained for multiplication, we cancel any common factors contained in the divisor and the numerator. Hence, if the numerator be exactly divisible by the divisor, we may divide a fraction by a whole number by dividing the numerator by that number.

EXAMPLE 1:

3 27 2̸7̸ 3 -- ÷ 18 = ------- = -- _Ans._ 31 31 × 1̸8̸ 62 2

EXAMPLE 2:

36 4 -- ÷ 9 = -- _Ans._ 41 41

(ii) When the divisor is a fraction.

In the operation 24 ÷ 3, we have to find the number which, when multiplied by 3, will give 24. Similarly, to find the value of 3/7 ÷ 5/9 we have to find the fraction which, when multiplied by 5/9, will give 3/7.

3 × 9 But ----- is the fraction which gives 3/7 when multiplied by 5/9. 7 × 5

3 5 3 × 9 Therefore, - ÷ - = -----. 7 9 7 × 5

Hence, to divide by a fraction, invert the divisor and multiply.

As in multiplication, mixed numbers must first be reduced to improper fractions.

EXAMPLE 3: Divide 3-1/14 by 5-5/42.

3 1 5 43 215 4̸3̸ 4̸2̸ 3 3-- ÷ 5-- = -- ÷ --- = -- × --- = - _Ans._ 14 42 14 42 1̸4̸ 2̸1̸5̸ 5 5

DECIMAL FRACTIONS

Differ in form from common fractions, in not having a written denominator; and from whole numbers, by having the decimal point (.) prefixed; which also separates the integral part from the decimal. The word decimal is derived from the Latin word _decem_, which signifies ten. The denominator of a decimal is always 10, or some power of 10, as 100, 1000, etc.

A Complex Decimal is a decimal with a common fraction at the right, as, .12-1/2.

A Mixed Decimal is a whole number with a decimal fraction to its right, as, 34.5.

The denominations of United States money are based on the decimal system--the dollar occupying the unit’s place, the dime the tenth’s place, the cent the hundredth’s place, and the mill the thousandth’s place.

_The rules given for addition, subtraction, and so on, also apply to decimals._

ADDITION IN DECIMALS

EXAMPLE: 27.295 + .0287 + 591.68 + 9.1846.

27.295 .0287 591.68 9.1846 -------- 628.1883 _Ans._

Write the numbers so that the same powers of 10 come under one another, or, what is the same thing, write the numbers so that the decimal points come under one another. Then, adding the ten-thousandths first, 6, 13, carry 1, etc.

SUBTRACTION IN DECIMALS

EXAMPLE: Subtract .07295 from 21.651.

21.651 .07295 -------- 21.57805 _Ans._

Write the first number under the second, so that the point comes under the point. Remember that we may consider there are 0’s above the 9 and 5, since in 21.651 there are no ten-thousandths and no hundred-thousandths.

Say, mentally 5 and 5 make 10, carry 1. 10 and 0 make 10, carry 1. 3 and 8 make 11, carry 1, etc.

MULTIPLICATION IN DECIMALS

RULE.--Multiply as in whole numbers, and point off from the right of the product as many places as there are decimal places in both multiplier and multiplicand--prefixing ciphers if necessary.

EXAMPLE 1: Multiply 87.432 by 564.

87.432 564 --------- 43716.0 5245.92 349.728 --------- 49311.648 _Ans._

Place the multiplier so that its unit’s digit comes under the right-hand digit of the multiplicand. Then place the first figure of each product underneath the multiplying digit. The decimal point of the answer will then be directly under the decimal point of the multiplicand.

EXAMPLE 2: Multiply 31.56 by 5.49.

31.56 5.49 -------- 157.80 12.624 2.8404 --------- 173.2644 _Ans._

As before, place the unit’s figure of the multiplier--that is, the 5--under the right-hand digit of 31.56, and proceed as above.

Note.--The number of decimal places in the product will always be equal to the sum of the number of decimal places in the multiplier and the multiplicand. Thus, in Example 2, there are two places of decimals (_i.e._ two figures to the right of the point) in 31.56, and two places of decimals in 5.49; and we found 2 + 2 = 4 places in the product 173.2644.

To multiply a decimal by 10, 100, etc.

Rule.--Remove the (.) as many places to the right as there are ciphers in the multiplier.

Work: 8.75 × 10 = 87.5 8.75 × 100 = 875. 8.75 × 1000 = 8750.

DIVISION OF DECIMALS

RULE.--Divide as in whole numbers, annexing ciphers to the dividend, if necessary; then point off from the right of the quotient as many places as the decimal places in the dividend exceed those in the divisor--prefixing ciphers if necessary.

(a) Division of a decimal by a whole number.

EXAMPLE 1: Divide 18.2754 by 4.

4)18.2754 -------- 4.56885

We divide 4 into 18 (units) and have 4 (units) quotient and 3 units remainder. Since the 4 is the unit’s figure of the quotient, we write the decimal point immediately after it. Then, the 2 units remainder and the 2 tenths of the dividend make 22 tenths to be divided by 4, and so on. Having reached the 4 (ten-thousandths) of the dividend, we find 8 (ten-thousandths) quotient and 2 remainder. This remainder is 20 hundred-thousandths, which when divided by 4 gives 5 (hundred-thousandths) and no further remainder.

EXAMPLE 2: Divide 18.2758 by 11.

11)18.2758 -------- 1.66143636

Here we find the digits 3, 6 repeated indefinitely in the quotient. Decimals of this sort will be fully considered later.

EXAMPLE 3: Divide 354.43 by 184.

184)354.43(1.92625 _Ans._ ---- 1704 ---- 483 --- 1150[16] ---- 460 --- 920 ---

Here we find the first figure of the quotient is obtained by dividing 184 into 354 units. Having now reached the decimal point in the dividend we also put the decimal point in the answer, and go on as before.

[16] At this stage there is a remainder 115 hundredths. We bring down 0 from the dividend, and obtain 1150 thousandths, etc.

(b) Division of a decimal.

EXAMPLE 4: Divide 10.6603 by 7.85.

Thus:

785)1066.03(1.358 _Ans._ ----- 2810 ---- 4553 ---- 6280 ----

Here 7.85 is 785 hundredths, and 10.6603 is 1066.03 hundredths; so that the required quotient is obtained by dividing 1066.03 by 785.

Therefore, to divide by a decimal, move the point as many places to the right as will make the divisor a whole number; move the point in the dividend the same number of places to the right. Then proceed as in Example 3.

EXAMPLE 5: Divide 176.4 by .00012.

12)17640000 --------- _Ans._ 1470000

Here, to make the divisor a whole number, we have to move the point 5 places. Therefore we also move the point 5 places to the right in the dividend, first writing enough 0’s after the 176.4 to enable us to do so.

To divide a decimal by 10, 100, etc.

RULE.--Remove the (.) as many places to the left as there are ciphers in the divisor.

WORK:

62.5 ÷ 10 = 6.25 62.5 ÷ 100 = .625 62.5 ÷ 1000 = .0625

Expression of decimal fractions as common fractions.

EXAMPLE: Express 5.375 as a common fraction.

.375 = 375 thousandths.

375 3 Therefore 5.375 = 5---- = 5- _Ans._ 1000 8

RULE.--_Take the digits of the decimal for numerator; for the denominator put down 1 followed by as many ciphers as there are digits in the decimal. Reduce this fraction to its lowest terms._

Expression of common fractions as decimals.

We have seen that a common fraction represents the quotient of the numerator divided by the denominator. Therefore, to convert a common fraction to a decimal fraction, we divide the numerator by the denominator.

EXAMPLE: Express 3/32 as a decimal.

4) 3.0 --------- 8) .75 --------- .09375 _Ans._

It will be found in many cases that there is always a remainder, so that the quotient can be continued indefinitely.

CIRCULATING DECIMALS

The learner has already discovered that some common fractions cannot be changed to exact decimal fractions, as--

1/3 =.33333 on to infinity. 2/3 =.66666 on to infinity. 7/33 =.212121, etc.

These decimals are known as _Circulates_, _Recurring_ or _Circulating_ decimals.

The part which recurs is called the _Repetend_.

This is marked by putting a dot over the first and last figures of it. For instance, if we write the 21 in the last case above, this way: 2̊1̊, it indicates that, if written out, the result would be 21212121, etc., on to infinity.

Where a circulating decimal occurs in work, it is best to reduce it to a common fraction. If need be, it may be expressed in the result as a circulate to any number of decimal places.

To change a pure circulate to a common fraction.

RULE.--Omit the (.) and write the figures of the repetend for the numerator, and as many 9’s for the denominator as there are places in the repetend.

EXAMPLES: Change the pure circulates .3̊, .2̊7̊, .1̊42857̊, to common fractions.

(3 1) .3̊, (- = -) _Ans._ 1/3. (9 3)

(27 3) .2̊7̊, (-- = --) _Ans._ 3/11 (99 11)

(142857 1) .1̊42857̊, (------ = -) _Ans._ 1/7 (999999 7)

To change a mixed circulate to a common fraction.

RULE.--From the whole decimal subtract the finite part, and make the remainder the numerator. For the denominator, write as many 9’s as there are figures in the repetend, and annex as many 0’s as there are finite places.

EXAMPLE: Change the mixed circulates .16̊ and .416̊ to common fractions.

15 1 16 - 1 = 15, --- = -. _Ans._ 1/6. 90 6

375 5 416 - 41 = 375, --- = -- _Ans._ 5/12. 900 12

To add, subtract, multiply and divide circulates, reduce them to common fractions, then apply the respective rules.

SHORT METHODS IN MERCHANDISING

When one of the numbers is an _aliquot part_ of 100, the process of multiplication and division can often be very much shortened, as shown below.

Find cost of 27 yards of goods at 16-2/3c ($1/6) per yard. At $1 per yard, 27 yards cost $27; at $1/6, (27 ÷ 6), $4-1/2. _Ans._ $4-1/2.

Find cost of a bale of cotton, 528 pounds at 8-1/3c ($1/12) per pound. At $1 per pound, 528 pounds cost $528; at $1/12 (528 ÷ 12) $44. _Ans._ $44.

Find cost of 1845 pounds of iron, at 3-1/3c ($1/30) per pound. Take 1/30 of 1845, since 3-1/3c is 1/30 of $1. (1845 ÷ 30 = 61-1/2). _Ans._ $61-1/2.

Find cost of 16 pounds of butter at 37-1/2c ($3/8) per pound. Here we take 3/8 of 16. Say 1/8 of 16 is 2, and 3/8 is (2 × 3) 6. Or say 3 times 16 is 48, and 1/8 of 48 is 6. _Ans._ $6.

Find cost of 17-1/2 bushels of apples at 75c ($3/4) per bushel. The shortest way to find 3/4 of $17.50 is to diminish it by 1/4 of itself.

4)17.50 at $1 ------ 4.37-1/2 at $1/4 13.12-1/2 at $3/4

_Ans._ $13.12-1/2.

At 6-1/4c per pound how much sugar will $5 buy? As 6-1/4c is 1/16 of $1, evidently each dollar will buy 16 pounds. _Ans._ 80 pounds.

In multiplying by a fraction, write the quantity in a line with the numerator and cancel common factors.

Find cost of 72 yards of carpet, at 87-1/2c ($7/8) a yard. Cancel 8, also 72 and write 9 instead. _Ans._ $63.

7 × 7̸2̸ 9 = 63 - 8̸

Of 28 pounds of coffee, at 18-3/4c ($3/16) per pound. Cancel 28 and 16, write 7 and 4. _Ans._ $5-1/4.

3 × 2̸8̸ 7 21 ---- = -- or 5-1/4 1̸6̸ 4 4

At 66-2/3c ($2/3) per bushel, how many bushel of wheat will $34 buy? _Ans._ 51 bushel.

3 × 3̸4̸ 17 = 51 - 2̸

In division, invert terms of fraction.

How much syrup, at 41-2/3c ($5/12) per gallon can be bought for $15? _Ans._ 36 gallons.

12 × 1̸5̸ 3 = 36 -- 5̸

TABLE OF ALIQUOT PARTS OF 100