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.
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.
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, and510of 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
1. A fraction’s value is the quotient obtained by dividing the numerator by the denominator.
62= 33is the value of62
23=2323is the value of23
2. Multiplying the denominator of a fraction divides the fraction by that number.
12× 4=1837× 3=32123× 9=227
3. Dividing the denominator of a fraction multiplies the fraction by that number.
38÷ 4=32109÷ 3=103310÷ 5=32
4. Multiplying the numerator of a fraction multiplies the fraction by that number.
23× 2=4319× 8=8958× 3=158
5. Dividing the numerator of a fraction divides the fraction by that number.
47÷ 2=271216÷ 12=11637÷ 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=1367× 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=12151827÷ 9÷ 9=23=1827
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.
Reduce8⁄12to its lowest terms.
Work:4 )8⁄12(2⁄3Ans.2⁄3
Four is the G. C. D. of 8 and 12; hence8⁄12÷ 4 =2⁄3.
Reduce35⁄56to its lowest terms.
Work:7 )35⁄56(5⁄8Ans.5⁄8
Seven is the G. C. D. of 35 and 56; hence35⁄56÷ 7 =5⁄8.
A fraction whose terms have no common divisor is in its lowest terms, as9⁄16.
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 in30⁄6?
Work:30÷ 6 = 5Ans.5.
There are as many units in 30 sixths as 6 is contained times in 30.
Reduce75⁄4to 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 183⁄4to 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.
Reduce1⁄2,2⁄3and3⁄4to their L. C. D.
Work:12=61223=81234=912
Ans.6⁄12,8⁄12and9⁄12.
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 change1⁄2to 12ths, say,1⁄2of 12 is 6, and write it over 12; to change2⁄3to 12ths, say2⁄3of 12 is 8, and write it over 12; to change3⁄4to 12ths, say,3⁄4of 12 is 9, and write it over 12.
Fractions must be reduced to a common denominator to be added or subtracted.
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.
Example1:Find the value of19+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= 14⁄9Ans.
In adding mixed numbers, first add the whole numbers, then the fractions, finally adding the two results.
Example2: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= 15310Ans.
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 45⁄21from 63⁄7.
637- 4521= 6 - 4 +37-521
637- 4521= 2 +9 - 521
637- 4521= 2 +421= 2421Ans.
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 74⁄15take 411⁄25.
7415- 41125= 7 - 4 +415-1125
7415- 41125= 3 +20 - 3375
7415- 41125= 2 +75 + 20 - 3375
7415- 41125= 2 +6275= 26275Ans.
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=139315Ans.
[15]Obtained by adding all the numerators with + before them, and then all those with - before them.
[15]Obtained by adding all the numerators with + before them, and then all those with - before them.
(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 means79+79+79+79,i.e.,289or7 × 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:Multiply19⁄46by 69.
1946× 69 =19 × 6946=19 × 32(cancelling 23), =572= 2812Ans.
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,3⁄5×4⁄11may be looked upon as doing to3⁄5what we do to the unit to obtain4⁄11.
Now, to obtain4⁄11from the unit, we must divide the unit into 11 equal parts and take 4 of them.
Therefore, to find the value of3⁄5×4⁄11we must divide3⁄5into 11 equal parts and take 4 of them.
But3⁄5=33⁄55=3⁄55× 11, so that, the eleventh part of3⁄5is3⁄55; and, if we take 4 of these parts, we get3⁄55× 4 or12⁄55.
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: Multiply22⁄91by13⁄77.
Theproduct =222×13917×777=249Ans.
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.
Givenexpression =3367×11279×535242=5518= 3118
(i) When the divisor is a whole number. Suppose we have to divide7⁄9by 4.
We know7⁄9=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. Hence7⁄9÷ 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.
Example1:
2731÷ 18 =32731×182=362Ans.
Example2:
3641÷ 9 =441Ans.
(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 of3⁄7÷5⁄9we have to find the fraction which, when multiplied by5⁄9, will give3⁄7.
But3 × 97 × 5is the fraction which gives3⁄7when multiplied by5⁄9. 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.
Example3: Divide 31⁄14by 55⁄42.
3114÷ 5542=4314÷21542=4314×3422155=35Ans.
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 worddecem, 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, .121⁄2.
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.
Example: 27.295 + .0287 + 591.68 + 9.1846.
27.295.0287591.689.1846628.1883Ans.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.
27.295.0287591.689.1846628.1883Ans.
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.
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.
Example: Subtract .07295 from 21.651.
21.651.0729521.57805Ans.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.
21.651.0729521.57805Ans.
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.
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.
Say, mentally10 and 0 make 10, carry 1.
Say, mentally3 and 8 make 11, carry 1, etc.
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.
Example1: Multiply 87.432 by 564.
87.43256443716.05245.92349.72849311.648Ans.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.
87.43256443716.05245.92349.72849311.648Ans.
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.
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.
Example2: Multiply 31.56 by 5.49.
31.565.49157.8012.6242.8404173.2644Ans.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.
31.565.49157.8012.6242.8404173.2644Ans.
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.
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.
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.
Example1: Divide 18.2754 by 4.