Fractions: equal parts, equivalent amounts and simplest form
Before you start: Count equal groups and understand multiplication and division.
What the two numbers mean
Choose one pizza as the whole. Cut it into 8 equal pieces and take 3: you have 3/8 of that pizza. The denominator 8 names the size of each piece; the numerator 3 counts how many of those pieces you have. Three uneven slices cannot automatically be called 3/8. Comparisons also require the same-sized whole: half a large pizza can be more food than half a small one.
Why equivalent fractions represent the same amount
Suppose you have 1/2 of a pizza. Cut each of the original two pieces into three equal smaller pieces. The whole now has 6 pieces, and your half contains 3, so 1/2 = 3/6. You changed the piece size and count together, without adding food.
The general rule is a/b = (a × k)/(b × k), where b and k are nonzero. Multiplying the top alone changes the amount: 1/2 becomes 3/2, which is three times as much. Multiplying both parts is equivalent to multiplying by k/k = 1.
Simplify by making larger equal pieces
To simplify 6/8, divide both parts by their common factor 2: (6 ÷ 2)/(8 ÷ 2) = 3/4. Numerator and denominator now have no common factor greater than 1. You can check by multiplying both by 2 to recover 6/8. Simplest form changes the representation, not the quantity.
Compare using equal-sized parts
Which is larger, 2/3 or 3/4? Rewrite both in twelfths: 2/3 = 8/12 and 3/4 = 9/12. Nine equal pieces are more than eight, so 3/4 is larger. Comparing only the numerators or only the denominators would not establish this.
Try it yourself
Rewrite 2/5 in tenths. Then simplify 9/12.
Show answer and explanation
Multiply both parts of 2/5 by 2 to get 4/10. Divide both parts of 9/12 by their greatest common factor, 3, to get 3/4. Each operation preserves the amount.
Use the activity to check your understanding
Use the slicer to change the number of equal pieces. Predict the numerator needed to keep the same shaded amount.