SHORTCUT ON FRACTIONS AND MIXED FRACTIONS.

✓ADDING TWO FRACTIONS WHOSE NUMERATORS ARE BOTH 1.

Rule: Write the sum of the denominators over the product of the denominators.

The numerator is the number over the fraction line and, of course, the denominator is the number under the fraction line.

To find the sum of (1/7)+ (1/12).

merely add the denominators,

 7 + 12 = 19 

and put the sum over the product of the denominators. 7 X 12 = 84 Thus 1/7 + 1/12 = 19/84 Answer It is always a good idea to reduce any fractional answer to its simplest form by canceling any common factors. In the example above, 19 and 84 have no common factors; therefore 19/84 is the simplest form of the fraction. 

✓FINDING THE DIFFERENCE BETWEEN TWOFRACTIONS WHOSE NUMERATORS ARE BOTH 1

Rule: Write the difference between the denominators over the product of the denominators.

Watch how a couple of simple short cuts can save a lot of work and time in solving an otherwise difficult problem.  

Subtract 1/87 from 1/83. The difference between the denominators is 4. The product of the denonunators is 

 83 x 87 = 7,221 The answer is 4/7,221. A fractional answer should always be tested to see if it can be reduced to a simpler form by cancelling common factors in the numerator and denominator. In the case above, it is obvious that the only factors of 4 are 2 and 4. But since 7,221 is an odd number, it does not contain either a 2 or 4, so that 4/7,221 is the simplest form of the fraction.  

✓MULTIPLYING BY 3/4.

Rule: Divide the given number by 4 and subtract the result from the given number.

Multiply 8,924 by 3/4. Divide 8, 924 by 4 and subtract the result from the original number. 4) 8 924 Given number     -2,231 Quotient      6,693 Answer 

✓MULTIPLYING BY 2-1/2. (Two and one by two)

Rule: Starting from the first digit of the given number, double each digit and add one-half the given digit. Ignore any fractions. Add any tens digits to the previous answer digit. Add 5 if the preceding digit of the given number is odd. Affix } to the answer if the given number is odd.

Use this Rule when the given number is an integer or a decimal number. Mixed numbers and fractions are difficult to handle with this method. This rule will be used in the example: 517,849 x 2-1/2.

Double the 5 and add one-half of itself to obtain the first answer digit. 5 + 5 = 10; 10 + 5/2 =12 (The fraction is ignored.) Record 12 as the first two answer digits. Double 1. One-half of 1 is 1, which is ignored. But since the preceding given digit is odd, add 5 to the result of this step. 2 + 0 = 2, 

2 + 5 =7 Record the 7 as the next answer digit. 127 The next given digit is 7. 7+7= 14; 14+7/2= 17 The previous given digit, 1, is odd; therefore add 5. 

17 + 5 = 22 Record the units digit, 2, as an answer digit. Carry the tens digit, also 2, and add it to the previous answer digit. The answer thus far is 1292 Continue this process with the balance of the given digits. 8 + 8 = 16; 16 + 4 = 20; 20 + 5 = 25 Record 5; add the 2 to the previous answer digit. The answer at this point is 12945 The 4 is next. 8; 8 + 2 = 10 Record O; add I to previous answer digit. 129460 Next is 9. 18; 18 +9/2 = 22 Record 2; add 2 to the previous answer digit. 1,294,622 This is the last digit of the given number. Since the given number is odd, affix to the answer. 1,294, 622-1/2 Answer 

✓MULTIPLYING BY 7-1/2.

Rule: Move the decimal point one place to the right,divide by 4, and subtract the quotient from the number first divided.

Since difficulties are sometimes encountered when moving the decimal point of a fraction or mixed number,limit the use of this rule to integers and decimal numbers.

Multiply 63 by 7-1/2. Move the decimal point one place to the right. 63. 0→becomes 630. Divide by 4 and subtract the quotient from 630. 630 ÷ 4 = 157-1/2 630 -157-1/2= 472-1/2. Thus 63 × 7-1/2=  472-1/2 Answer. 

✓MULTIPLYING BY 12-1/2.

Rule : Move the decimal point two places to the right and divide by 8. Use this rule on whole numbers and decimal numbers only. Odd whole numbers will always end in 1/2,  i.e., the remainder after dividing by 8 will be 4. As an example : Multiply 631 by 129 · Move the decimal point two places to the right. 631. 00 becomes 63, 100. Divide by 8. 63, 100 ÷ 8 = 7, 887-1/2 Therefore 631 × 12-1/2 =7, 887-1/2 Answer. 

✓MULTIPLYING TWO MIXED NUMBERS WHOSE WHOLE NUMBERS ARE THE SAME AND WHOSE FRACTIONS ADD TO 1.

Rule : Multiply the whole number by one more than itself. Afíix the product of the fractions.M

Multiply9-4/9 by 9-5/9.

Multiply the whole number by one more than itself. 9 × 10 = 90 Affix the product of the fractions. 4/9 × 5/9 = 20/81 9-4/9 × 9-5/9 = 90-20/81 Answer. 

✓MULTIPLYING TWO MIXED NUATBERS WHEN THE  DIFFERENCE BETWEEN THE WHOLE NUMBERS IS 1 AND THE SUM OF THE FRACTIONS IS 1

Rule: Increase the Larger of the whole numbers by one and multiply by the other whole number. Square the fraction of the larger number and subtract the square from 1. Affix the result to the product obtained in the first step.

For the sum of two fractions to be equal to 1, their denominators must be the same (at least when both fractions are written their simplest form) and the sum of the numerators must equal the denominator. Multiply 15-3/4 by 14-1/4. Increase the larger whole number by one and multiply by the smaller whole number. 15 + 1 = 16 ; 16 × 14 = 224 Square the fraction of the larger number 3/4 × 3/4 = 9/16 Subtract the result from 1. 1 - 9/16 =  7/1 6 Affix the result to the product obtained in the first step. 224-7/16 Answer.

✓SQUARING A NUMBER ENDING IN 1/2.

Rule: Multiply the whole-number part of the given number by one more than itself and affix 1/4.

Naturally the ease with which the whole number is multiplied by one more than itself will determine when this rule is used. Often other rule methods can be applied to reduce the job of multiplying the whole numbers. 

Square 87-1/2 Multiply the whole number, 87, by one more than itself  87 x 88 = 7,656 Affix 1/4. 7, 656-1/4 Answer. 

✓DIVIDING BY 2-1/2

Rule: Move the decimal point one place to the left and multiply by 4.

If enough zeros are added to the right of the decimal point of the given number, there Will never be a fractional remainder left after dividing by 4. Divide 87.6 by Move the decimal point one place to the left. 87.6 becomes 8.76 Multiply by 4 (Short Cut 10).0 8.76 x 4 = 35.04 Therefore 87.6 ÷ 2-1/2 = 35.04 Answer 

✓DIVIDING BY 12-1/2.

Rule: Move the decimal point two places to the left and multiply by 8.

Use this short cut on integers and decimal numbers. Fractions and mixed numbers sometimes present problems in moving their decimal point. Example:  Divide 57,813 by 12-1/2. Move the decimal point two places to the left.

 57, 813. becomes 578,13 Multiply by 8. 578.13 x 8 = 4,625.04 Therefore 57,813 ÷ 12-1/2 =  4,625.04 Answer The answer is a decimal number. This will always be true (although, of course, the decimal portion can be zero at times). 

✓DIVIDING BY 33-1/3. Rule: Multiply the given number by 3 and move the decimal point two places to the left.

Divide 83 by 33-1/3. Multiply by 3 83 x 3 = 249 Move the decimal point two places to the left. 249.0 becomes 2.49 Thus 83 ÷ 33-1/3 = 2.49 Answer