Fraction × whole number

1Summary 重點摘要

25 × 3 = 65. Numerator × integer, denominator stays.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Part 1: What a fraction of an amount means14 of 12 sweets means sharing 12 sweets into 4 equal parts and taking one part, the basic meaning of a fraction of an amount. 12 ÷ 4 = 3, so 14 of 12 is 3 sweets. Here a fraction describes the action of sharing equally and then taking some parts. Take care that the sharing must be equal; if the parts are unequal, each part differs in size, one part has no fixed amount, and the meaning of the fraction collapses, so equal sharing always comes first.
2. Part 2: Unit fractions times a whole14 × 12 has two routes, and both reach the same answer. Route one divides directly: 12 ÷ 4 = 3. Route two uses the formula: numerator 1 × 12 = 12, denominator 4 unchanged, giving 124 = 3. Why does the formula work? 14 × 12 gathers twelve 14 pieces together, and after simplifying it arrives at the same place as direct division. Direct division is usually quicker, especially with small numbers. Take care to choose one route only; mixing the two in one line only tangles the working.
3. Part 3: Numerators bigger than 125 × 3 means three lots of 25; when the numerator is not 1, the formula is the most direct: numerator 2 × 3 = 6 with denominator 5 unchanged, giving 65. Since 65 is more than 1, it becomes the mixed number 1 15. Remember: numerator times the whole, denominator stays unchanged. Why does the denominator hold? The size of each part has not changed; only the number of parts taken has. Take care to multiply the numerator only; multiplying the denominator too changes the size of every part and gives a completely different answer.
4. Part 4: Divide first for smaller numbersFor 34 × 20, the formula gives (3 × 20)/4 = 604 = 15. A steadier way divides first: 20 ÷ 4 = 5, then 5 × 3 = 15. Why divide first? It keeps the numbers small and mistakes away, especially when the denominator is large. The 5 found first has meaning too: 14 of 20 is 5, and taking 3 such parts gives 15, which fits the question exactly. Take care that dividing first assumes the whole number divides exactly by the denominator; when it does not, use the formula, multiply first and divide after.
5. Part 5: Check the form of the answer23 × 4 = 83, and 83 is an improper fraction with the top bigger than the bottom, so change it into the mixed number 2 23. If top and bottom share a factor, simplify as well. Only after this final check of form is the work complete. The usual error is handing in 83 as it stands, without the mixed form; when the question asks for simplest form, an improper or unsimplified fraction does not count as finished, so tidy the form before handing in.
6. Part 6: Word problemA ribbon is 24 cm long and 34 of it is used; how many centimetres is that? 24 ÷ 4 = 6, then 6 × 3 = 18, so 18 cm is used. Left over is 24 - 18 = 6 cm, exactly 14 of the whole length, which serves as a check. To find a fraction of an amount, multiply, or divide first and then multiply. Take care that used plus left must return the whole length, 18 + 6 = 24; without this last check, a slip in numerator or denominator may pass unseen, and the check is the final line of defence in word problems.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• A quarter of 12 is written 14 × 12
• Fraction × whole: numerator × whole, denominator unchanged
• Easier route: divide first, then multiply, e.g. 34 × 20 → 5 × 3 = 15
• Turn improper answers into mixed numbers and simplify
• Word problems: divide for parts, multiply for how many parts taken