Mathematics I · No calculator

Operations with Fractions and Mixed Numbers

Four operations, one calm method each. By the end you'll add, subtract, multiply, and divide fractions by hand without flinching.

Fractions feel intimidating, but here's the good news: each operation has its own simple recipe, and you only ever use one recipe at a time. Multiplying is the easiest. Adding takes one extra setup step. Dividing is just multiplying with a tiny flip. Once you know which recipe to reach for, the fear melts away.

We'll start with the two skills that power everything else, then walk through all four operations with fully worked examples. No calculator needed, just a pen and a little patience.

First, the two parts of a fraction

Every fraction is written as top over bottom:

  • The numerator is the top number. It tells you how many parts you have.
  • The denominator is the bottom number. It tells you how many equal parts make a whole.
  • A mixed number is a whole number next to a fraction, like 2 1/3 (two and one third).

Power skill #1: simplifying to lowest terms

To simplify a fraction, divide the top and the bottom by the same number until nothing divides them evenly anymore. The value doesn't change, it just gets tidier.

812 = 8 ÷ 412 ÷ 4 = 23

Both 8 and 12 divide by 4, so 8/12 simplifies to 2/3.

A good habit: always simplify your final answer. If the top and bottom are both even, start by dividing both by 2.

Power skill #2: converting mixed numbers and improper fractions

An improper fraction has a top bigger than its bottom (like 7/3). Before you multiply or divide mixed numbers, turn them into improper fractions first. The recipe is multiply, then add.

213 = (2 × 3) + 13 = 73

Multiply 2 × 3 = 6, add the 1 to get 7, and keep the same bottom (3).

To go back the other way, divide the top by the bottom. The whole number is the quotient, and the remainder becomes the new top:

73 = 7 ÷ 3 = 2 remainder 1 = 213

Multiplying fractions: straight across

This is the easiest one, so we'll start here. To multiply, you do not need a common denominator. Just multiply the tops together and the bottoms together, then simplify.

  • Tops: multiply the two numerators.
  • Bottoms: multiply the two denominators.
  • Simplify: reduce the result to lowest terms.
23 × 45 = 2 × 43 × 5 = 815

8 and 15 share no common factor, so 8/15 is already in lowest terms.

Answer: 2/3 × 4/5 = 8/15. With mixed numbers, convert to improper fractions first, then multiply straight across.

Dividing fractions: keep, change, flip

Dividing is just multiplying in disguise. Use the keep-change-flip trick:

  • Keep the first fraction exactly as it is.
  • Change the division sign to a multiplication sign.
  • Flip the second fraction upside down (this is called its reciprocal).
34 ÷ 25 = 34 × 52(keep, change, flip)
3 × 54 × 2 = 158 = 178
Answer: 3/4 ÷ 2/5 = 15/8 = 1 7/8. Only the second fraction flips. The first one stays put.

Adding and subtracting: you need a common denominator

You can only add or subtract fractions when the bottoms match. If they already match, just add (or subtract) the tops and keep the bottom the same:

27 + 17 = 2 + 17 = 37

When the bottoms are different, rewrite each fraction so they share a common denominator. The quickest reliable choice is to multiply the two bottoms together.

  • Find a common bottom: for 1/4 and 1/6, you can use 4 × 6 = 24.
  • Rebuild each fraction: multiply each top by whatever you multiplied its bottom by.
  • Add the tops: keep the common bottom, then simplify.
14 = 1 × 64 × 6 = 624
16 = 1 × 46 × 4 = 424
624 + 424 = 6 + 424 = 1024 = 512
Answer: 1/4 + 1/6 = 10/24 = 5/12. Subtraction works exactly the same way; just subtract the tops instead of adding them.

Worked example: adding mixed numbers

For mixed numbers, the safest approach is to turn each one into an improper fraction, give them a common bottom, add, then convert back to a mixed number at the end. Let's add 2 1/2 + 1 1/3.

Step 1, make them improper:

212 = 52113 = 43

Step 2, common bottom of 6:

52 = 15643 = 86

Step 3, add the tops:

156 + 86 = 236

Step 4, back to a mixed number:

236 = 23 ÷ 6 = 3 remainder 5 = 356
Answer: 2 1/2 + 1 1/3 = 23/6 = 3 5/6.

Quick tips that prevent mistakes

  • Match the recipe to the sign. Plus or minus means "get a common bottom first." Times means "straight across." Divide means "keep, change, flip."
  • Convert mixed numbers early when multiplying or dividing. Trying to multiply mixed numbers directly is a common trap.
  • Only flip the second fraction when dividing. Flipping both, or flipping the first, gives the wrong answer.
  • Always simplify at the end and turn a top-heavy answer back into a mixed number if the question started with mixed numbers.

Your turn: practice problems

Work each one by hand and simplify your answers. No peeking until you've tried.

  1. 3/5 × 2/3
  2. 5/6 ÷ 1/2
  3. 2/3 + 1/4
  4. 3 1/2 − 1 1/4
Tap to reveal the answers
  • 1. Straight across: (3 × 2)/(5 × 3) = 6/15 = 2/5.
  • 2. Keep-change-flip: 5/6 × 2/1 = 10/6 = 5/3 = 1 2/3.
  • 3. Common bottom 12: 8/12 + 3/12 = 11/12.
  • 4. Subtract the fraction parts (1/2 − 1/4 = 1/4) and the whole parts (3 − 1 = 2): 2 1/4.

Why this matters for the CAEC

Mathematics Part I is the no-calculator section, so confident fraction work by hand is essential. Fractions show up in recipes, measurements, money, and rates, and the four operations here are the foundation for decimals, percentages, and word problems later.

Want more practice like this? Our CAEC math guide and the CAEC Ready Workbook are packed with worked examples and practice questions, or start with a free math sample to test yourself.

Disclaimer

This article is a general math tutorial for study purposes. CAEC Ready is an independent study resource and is not affiliated with or endorsed by any government, ministry of education, or official CAEC testing provider.

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