Simple Fractions
A very good method of learning the combinations in small fractions is by the use of paper or cardboard disks.
Cut out a large number of them, and, in order to avoid trouble later on, it might be better to have the disks all of one size—about 4 inches in diameter.
Learning the Fraction 1 ⁄ 2 With Disks
Explanation:—Take a circular disk and cut it into two equal parts. Then proceed in this manner: What is this part called? What is other part called? How many halves in the whole circle? One-half and one-half are what? One-half taken away from one leaves what? If I take a half two times, what do I get? How many halves in a whole?
Now I will write these—
12 + 12 =
1 ÷ 2 =
1 less 12 =
1 - 12 =
2 × 12 =
1 divided by 12 =
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
12, 24, 36, 48, and 510 of an apple.
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,
38, 271⁄2, 511, 1920, 2020, 1816, 115, 31⁄2, 162⁄3
Principles of Fractions
1. A fraction’s value is the quotient obtained by dividing the numerator by the denominator.
62 = 3 3 is the value of 62
23 = 23 23 is the value of 23
2. Multiplying the denominator of a fraction divides the fraction by that number.
12 × 4 = 18 37 × 3 = 321 23 × 9 = 227
3. Dividing the denominator of a fraction multiplies the fraction by that number.
38 ÷ 4 = 32 109 ÷ 3 = 103 310 ÷ 5 = 32
4. Multiplying the numerator of a fraction multiplies the fraction by that number.
23 × 2 = 43 19 × 8 = 89 58 × 3 = 158
5. Dividing the numerator of a fraction divides the fraction by that number.
47 ÷ 2 = 27 1216 ÷ 12 = 116 37 ÷ 3 = 17
6. Multiplying both numerator and denominator of a fraction by the same number does not change the value of the fraction.
13 × 3 × 3 = 39 = 13 67 × 2 × 2 = 1214 = 67
7. Dividing both numerator and denominator of a fraction by the same number does not change the value of the fraction.
1215 ÷ 3 ÷ 3 = 45 = 1215 1827 ÷ 9 ÷ 9 = 23 = 1827
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 ( 2⁄3Ans. 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 ( 5⁄8Ans. 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 = 5Ans. 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 + 3Ans. 183⁄4.
In 75 fourths there are 18 units, and 3 fourths over, which equals 183⁄4.
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 183⁄4 to an improper fraction.
Work: 18 × 4 = 7272⁄4 + 3⁄4 = 75⁄4Ans. 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: 12 = 612 23 = 812 34 = 912
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: 17 + 47 + 57 = 1 + 4 + 57 = 107 = 137
If the fractions have different denominators, we must first express them as equivalent fractions with the same denominator.
Example 1: Find the value of 19 + 37 + 521 + 23
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: 19 + 37 + 521 + 23 = 7 + 27 + 15 + 4263 = 91⁄63 = 128⁄63 = 14⁄9 Ans.
In adding mixed numbers, first add the whole numbers, then the fractions, finally adding the two results.
Example 2: Add together 31⁄8 + 7⁄24 + 711⁄15 + 43⁄20. Given expression:
= 3 + 7 + 4 + 18 + 724 + 1115 + 320
= 14 + 15 + 35 + 88 + 18120
= 14 + 156120 = 14 + 136120 = 15310 Ans.
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 45⁄21 from 63⁄7.
637 - 4521 = 6 - 4 + 37 - 521
637 - 4521 = 2 + 9 - 521
637 - 4521 = 2 + 421 = 2421 Ans.
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 74⁄15 take 411⁄25.
7415 - 41125 = 7 - 4 + 415 - 1125
7415 - 41125 = 3 + 20 - 3375
7415 - 41125 = 2 + 75 + 20 - 3375
7415 - 41125 = 2 + 6275 = 26275 Ans.
Example 3: Simplify 32⁄9 + 45⁄7 - 513⁄21 + 2⁄35 - 114⁄15. Given expression:
= 3 + 4 - 5 - 1 + 29 + 57 - 1321 + 235 - 1415
= 1 + 70 + 225 - 195 +18 - 294315
= 1 + 313 - 489[15]315
= 628 - 489315 = 139315 Ans.
[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:
79 × 4 means 79 + 79 + 79 + 79, i.e., 289 or 7 × 49
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.
1946 × 69 = 19 × 6946 = 19 × 32 (cancelling 23), = 572 = 2812 Ans.
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.
Thus, 35 × 411 = 1255. Now 12 = 3 × 4, and 55 = 5 × 11.
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.
The product = 2 22 × 13 91 7 × 77 7 = 249 Ans.
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 51⁄7 × 11⁄27 × 111⁄24.
Given expression = 3 36 7 × 11 27 9 × 5 35 24 2 = 55 18 = 3 1 18
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:
2731 ÷ 18 = 3 27 31 × 18 2 = 3 62 Ans.
Example 2:
3641 ÷ 9 = 441 Ans.
(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.
But 3 × 97 × 5 is the fraction which gives 3⁄7 when multiplied by 5⁄9. Therefore, 37 ÷ 59 = 3 × 97 × 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 31⁄14 by 55⁄42.
3114 ÷ 5542 = 4314 ÷ 21542 = 4314 × 3 42 215 5 = 35 Ans.