What are some mental math shortcuts for fractions and decimals?

1 Comparing fractions

How can you tell which fraction is bigger or smaller without converting them to decimals? One way is to cross-multiply the numerators and denominators and compare the results. For example, to compare 3/4 and 5/8, you can multiply 3 by 8 and 4 by 5. You get 24 and 20, respectively. Since 24 is greater than 20, 3/4 is greater than 5/8. Another way is to find a common denominator and compare the numerators. For example, to compare 2/3 and 3/5, you can multiply both fractions by 15, the least common multiple of 3 and 5. You get 10/15 and 9/15, respectively. Since 10 is greater than 9, 2/3 is greater than 3/5.

2 Converting fractions to decimals

How can you convert a fraction to a decimal without dividing the numerator by the denominator? One way is to use equivalent fractions that have a power of 10 as the denominator. For example, to convert 3/4 to a decimal, you can multiply both the numerator and the denominator by 25, which is a power of 10. You get 75/100, which is the same as 0.75. Another way is to use long division and place a decimal point after the numerator. For example, to convert 7/12 to a decimal, you can divide 7 by 12 and place a decimal point after 7. You get 0.5833.

3 Converting decimals to fractions

How can you convert a decimal to a fraction without using a calculator or a table? One way is to count the number of digits after the decimal point and use that as the power of 10 for the denominator. For example, to convert 0.25 to a fraction, you can count that there are two digits after the decimal point and use 100 as the denominator. You get 25/100, which can be simplified to 1/4. Another way is to use the place value of the decimal and write it as a fraction over the corresponding power of 10. For example, to convert 0.375 to a fraction, you can write it as 375 thousandths and use 1000 as the denominator. You get 375/1000, which can be simplified to 3/8.

4 Simplifying fractions

How can you simplify a fraction without finding the greatest common factor of the numerator and the denominator? One way is to use prime factorization and cancel out the common factors. For example, to simplify 24/36, you can write both numbers as products of prime factors: 24 = 2 x 2 x 2 x 3 and 36 = 2 x 2 x 3 x 3. Then you can cancel out the common factors of 2 and 3. You get 2/3. Another way is to use divisibility rules and divide both numbers by the same factor until they are relatively prime. For example, to simplify 24/36, you can divide both numbers by 2, then by 3, then by 2 again. You get 2/3.

5 Adding and subtracting fractions

How can you add or subtract fractions without finding the least common multiple of the denominators? One way is to use cross-multiplication and addition or subtraction. For example, to add 2/5 and 3/7, you can multiply 2 by 7 and 3 by 5 and add the results. You get 14 + 15 = 29. Then you can multiply the denominators and use that as the new denominator. You get 29/35. Another way is to use equivalent fractions and addition or subtraction. For example, to add 2/5 and 3/7, you can multiply both fractions by 35, the least common multiple of 5 and 7. You get 14/35 and 15/35, respectively. Then you can add the numerators and use the same denominator. You get 29/35.

6 Multiplying and dividing fractions

How can you multiply or divide fractions without using a calculator or a table? One way is to use cross-simplification and multiplication or division. For example, to multiply 3/4 and 4/9, you can simplify the numerator of one fraction and the denominator of the other by dividing them by their common factor of 4. You get 3/1 and 1/9, respectively. Then you can multiply the numerators and the denominators. You get 3/9, which can be simplified to 1/3. Another way is to use reciprocal and multiplication or division. For example, to divide 3/4 by 4/9, you can flip the second fraction and multiply it by the first. You get 3/4 x 9/4, which can be simplified to 27/16.

7 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?