1 · What the two numbers mean
The bottom (denominator) says how many equal parts make one whole. The top (numerator) says how many you have. 3/4: 4 equal parts, 3 shaded.
The same fraction can be a shape, a set, or a point on a number line.
2 · Equivalent fractions
Cut every piece of 1/2 in two and you get 2/4. Same amount, new name: 1/2 = 2/4 = 4/8.
Multiply top and bottom by the same number and the fraction does not change: 2/3 = 4/6 = 6/9.
3 · Simplest form
Go the other way: divide top and bottom by the same number. 6/8 ÷ 2 = 3/4. When nothing divides both, the fraction is in simplest form.
4/12: both divide by 4, so 4/12 = 1/3.
4 · Comparing: same bottom, same top
Same bottom: more pieces wins. 3/5 > 2/5.
Same top: the smaller bottom wins, because the pieces are bigger. 1/3 > 1/5.
5 · Comparing: different bottoms
2/3 or 3/4? Rename both to twelfths: 2/3 = 8/12 and 3/4 = 9/12. So 3/4 is greater.
Shortcut: compare to a half. 2/3 is a half plus 1/6; 3/4 is a half plus 1/4. A quarter is bigger than a sixth, so 3/4 wins.
6 · Fractions on the number line
Split the space between 0 and 1 into equal steps. On a line marked in fourths, the dot at the third mark is 3/4.
Past 1 the counting keeps going: 5/4 is one whole and one more quarter, which is 1 1/4.
7 · Adding and subtracting, same bottom
Same bottom? Add or subtract the tops and keep the bottom. 3/6 − 1/6 = 2/6, which simplifies to 1/3.
2/5 + 2/5 = 4/5. 3/5 − 2/5 = 1/5. Never add the bottoms.
8 · Making a whole
All the parts together make 1: 4/4 = 1, 6/6 = 1. So 1 − 1/4 = 3/4: take one quarter away from four quarters.
2/5 + ? = 1: you need 3 more fifths. ? = 3/5.
9 · Mixed numbers and improper fractions
7/4 is more than 1: four quarters make a whole, and 3 are left. So 7/4 = 1 3/4, a mixed number.
Back the other way: 2 1/3 = 2 wholes (6/3) plus 1/3 = 7/3.
10 · A fraction of a number
3/5 of 75: split 75 into 5 equal parts (75 ÷ 5 = 15), then take 3 of them (3 × 15 = 45). The picture shows the same idea with 15: 5 parts of 3, take 3 parts, that is 9.
Divide by the bottom, multiply by the top. 1/4 of 32 = 8. 2/3 of 27 = 18.
11 · Working backwards
4/5 of a number is 24. What is the number? If 4 parts are 24, one part is 24 ÷ 4 = 6. The whole is 5 parts: 5 × 6 = 30.
Divide by the top to find one part, then multiply by the bottom.
12 · Fractions in stories
"A group has 75 students. 3/5 are chosen. How many?" 75 ÷ 5 = 15, 3 × 15 = 45.
"Mia ate 2/8 of a pizza and Leo ate 3/8. How much is left?" 2/8 + 3/8 = 5/8 eaten, so 8/8 − 5/8 = 3/8 left. You are ready for the test! 🎉
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