More Than a Half

The halfway line

Draw a dashed line exactly halfway along a bar. However many equal parts the bar is cut into, the line stays in the same place, because half of the bar is always the same length.

Now shade parts from the left, one at a time. The shading stops before the line, ends right on it, or crosses it. That tells you whether the fraction is less than a half, exactly a half, or more than a half.

$2 \times 1 = 2 < 5, \frac{1}{5} < \frac{1}{2}$

2 × 1 = 2 < 5, so 1/5 is less than a half: the shading stops before the half line

Shade the fewest parts that make more than a half

The bar is cut into 5 parts and 1 part is shaded. Shade one more part at a time: 2 fifths stop short of the dashed line, and 3 fifths cross it.

Double the top

The decision can be made without a drawing. Half of 5 parts is two and a half parts, so 3 parts are more than a half and 2 parts are less. The quick check is to double the top number and compare it with the bottom number: 2 × 3 = 6, and 6 is more than 5, so 3/5 is more than a half.

If double the top is less than the bottom, the fraction is less than a half: 2 × 3 = 6 is less than 8, so 3/8 is less than a half. If double the top is equal to the bottom, the fraction is exactly a half: 2 × 2 = 4, so 2/4 is a half, and so are 3/6, 4/8 and 5/10.

Only an even bottom number can make exactly a half. A bar cut into 5 parts has no whole number of parts that ends on the line: 2 parts stop short of it and 3 parts cross it.

The dashed line is in the same place on all three bars. 3/8 stops before it, 2/4 ends on it and 3/5 crosses it.

Worked example: Three Jars Sorted as Less Than, Equal to or More Than a Half

Question Three jars are the same size. Jar A is 310 full, jar B is 48 full and jar C is 712 full. (a) Which jar is exactly half full? (b) Which jar is less than half full, and which jar is more than half full?

1.Jar A is cut into 10 parts. Half of 10 is 5, so 510 is a half. Jar A has only 3 parts, so it is less than half full.

Half of 10 is 5. Jar A has only 3 parts, so it is less than half full.
2.Jar B is cut into 8 parts. Half of 8 is 4, so 48 is exactly a half.

Half of 8 is 4, so 48 is exactly a half.
3.Jar C is cut into 12 parts. Half of 12 is 6, so 612 is a half. Jar C has 7 parts, so it is more than half full.

Half of 12 is 6. Jar C has 7 parts, so it is more than half full.
4.(a) Jar B is exactly half full.

(a) Jar B is exactly half full.
5.(b) Jar A is less than half full and jar C is more than half full.

(b) Jar A is less than half full and jar C is more than half full.

Answer: (a) Jar B; (b) jar A is less than half full and jar C is more than half full

Common mistakes

Comparing the top number with the wrong half. For jar C the half is 6 parts out of 12, not 5 parts or 4 parts. Each jar has its own half, found from its own bottom number.
Saying 48 is not a half because it does not look like 12. 4 is half of 8, so 48 and 12 are the same amount.

More comparing fractions problems, worked step by step →

Comparing two fractions with a half

A half also compares two fractions that share neither number. Take 3/7 and 5/9. Double 3 is 6, less than 7, so 3/7 is less than a half. Double 5 is 10, more than 9, so 5/9 is more than a half. A fraction above a half is bigger than one below it, so 5/9 is the bigger fraction.

Worked example: Every Numerator That Fits between a Half and One Whole

Question Nora folds a paper strip into 8 equal parts. She colors more than 12 of the strip, but she does not color the whole strip. (a) How many parts make exactly 12 of the strip? (b) How many parts could Nora have colored? Give every possible number.

1.(a) Half of 8 is 4, so 4 parts make exactly 12 of the strip: 48 = 12.

(a) Half of 8 is 4, so 4 parts make exactly 12 of the strip.
2.More than a half means more than 4 parts, so Nora colors at least 5 parts.

More than a half means at least 5 parts.
3.The whole strip is 8 parts, and 88 = 1. Less than the whole strip means fewer than 8 parts, so Nora colors at most 7 parts.

The whole strip is 8 parts, so Nora colors at most 7 parts.
4.(b) Nora could have colored 5, 6 or 7 parts. The fractions are 58, 68 and 78.

(b) Nora could have colored 5, 6 or 7 parts: 58, 68 or 78.

Answer: (a) 4 parts; (b) 5, 6 or 7 parts

Common mistakes

Including 4 parts. 48 is exactly a half, and the question says more than a half.
Including 8 parts. 88 is the whole strip, and Nora does not color the whole strip.

More comparing fractions problems, worked step by step →