Fractions and decimals without a calculator

NEIEP's study guide lists fractions, decimals and percentages among the EIAT math topics, and its examples (a decimal division and a mixed-number multiplication) are meant to be worked by hand. None of it is advanced — it is the arithmetic most of us did at school and have not touched since. This refresher covers the methods, with worked examples you can check yourself.

Fractions: the basics

A fraction is a numerator over a denominator: in 3/4, the numerator is 3 and the denominator is 4. A mixed number such as 2 1/2 is a whole number and a fraction together.

Converting a mixed number to an improper fraction: multiply the whole number by the denominator, add the numerator, and keep the denominator. 2 1/2 = (2 × 2 + 1)/2 = 5/2.

Simplifying: divide the top and bottom by the largest number that goes into both. 6/8 = 3/4.

Adding and subtracting fractions

You can only add or subtract fractions with the same denominator. Rewrite them over a common denominator first.

Example: 1 3/4 + 2 5/8

• Rewrite 3/4 as 6/8 so both use eighths: 1 6/8 + 2 5/8.
• Add the whole numbers and the fractions: 3 11/8.
• 11/8 is 1 3/8, so the answer is 4 3/8.

Subtracting with borrowing: 5 1/4 − 2 3/4. You cannot take 3/4 from 1/4, so borrow a whole: 5 1/4 = 4 5/4. Then 4 5/4 − 2 3/4 = 2 2/4 = 2 1/2.

Multiplying fractions

Convert mixed numbers to improper fractions, then multiply the tops together and the bottoms together.

Example: 3 1/2 × 2/3

• 3 1/2 = 7/2.
• 7/2 × 2/3 = 14/6.
• Simplify: 14/6 = 7/3 = 2 1/3.

Tip: cancel before multiplying where you can. Here the 2 on the bottom of 7/2 cancels with the 2 on the top of 2/3, giving 7/3 directly.

Dividing fractions

Flip the second fraction upside down and multiply.

Example: 2 1/4 ÷ 3/8

• 2 1/4 = 9/4.
• 9/4 ÷ 3/8 = 9/4 × 8/3 = 72/12 = 6.

Multiplying decimals

Ignore the decimal points, multiply as whole numbers, then put the point back: the answer has as many decimal places as the two numbers put together.

Example: 0.25 × 0.4. Multiply 25 × 4 = 100. There are two decimal places in 0.25 and one in 0.4, so three in total: 0.100 = 0.1.

Dividing by a decimal

Move the decimal point in both numbers the same number of places until the number you are dividing by is a whole number. Then divide normally.

Example: 12.6 ÷ 0.3. Move both points one place: 126 ÷ 3 = 42.

Example: 7.5 ÷ 0.25. Move both points two places: 750 ÷ 25 = 30.

Converting between fractions and decimals

A few conversions come up constantly, and knowing them saves time:

• 1/2 = 0.5, 1/4 = 0.25, 3/4 = 0.75
• 1/8 = 0.125, 3/8 = 0.375, 5/8 = 0.625, 7/8 = 0.875
• 1/5 = 0.2, 1/10 = 0.1

Eighths and sixteenths are also what you need to read a tape measure, a basic skill for tool knowledge.

Check your answer against the options

NEIEP's example math questions include a fifth option, "No answer". Once you have a result, check it against the four figures at the precision they are written in. If none matches, choose "No answer" rather than the closest figure.

Practice

Our EIAT math practice is built on exact fractions, so every answer and every wrong option is computed precisely, with "No answer" correct often enough that you have to check.