Dividing Mixed Numerals
Dividing Mixed Numerals builds on dividing fractions in NSW Year 7 maths (NESA outcome MA4‑FRC‑C‑01). Change both numbers to improper fractions, then multiply by the reciprocal of the second, so \(4\frac{1}{2} \div 1\frac{1}{2} = \frac{9}{2} \times \frac{2}{3} = 3\).
You will cancel common factors, change answers back to mixed numerals, estimate by rounding to whole numbers, find a missing factor, and test the claim that dividing always makes a number smaller. Cutting fencing wire into lengths with some left over and finding a car's average speed are real examples. It is part of Year 7 Fractions.
Every question with a fully worked solution.
- Dividing Mixed Numerals - Video - Dividing Mixed Numbers | Step by Step Watch
Theory
To divide mixed numerals, change them to improper fractions, then multiply by the reciprocal of the second number. This Year 7 maths page shows each step, how to estimate first, how to find the reciprocal of a mixed numeral and how to answer cutting and speed problems.
Step 1. Change to improper fractions: \(2\dfrac{2}{5} = \dfrac{2 \times 5 + 2}{5} = \dfrac{12}{5}\). A whole number goes over \(1\).
Step 2. Keep, change, flip: keep the first fraction, change \(\div\) to \(\times\), flip the second (use its reciprocal).
Step 3. Cancel common factors, then multiply tops and bottoms.
Step 4. Change back to a mixed numeral: \(\dfrac{7}{2} = 3\dfrac{1}{2}\).
Dividing by a number greater than \(1\) gives a smaller answer; dividing by a number between \(0\) and \(1\) gives a larger answer.
Improper fractions, then keep, change, flip:
How to divide mixed numerals
- Estimate by rounding to whole numbers.
- Change to improper fractions.
- Keep, change, flip, then cancel and multiply.
- Change back to a mixed numeral and check: answer \(\times\) divisor \(=\) first number.
(i) \(3\dfrac{1}{4}\)
(ii) \(1\dfrac{3}{5}\)
Improper fraction first, then flip: \(3\dfrac{1}{4} = \dfrac{13}{4}\) and \(1\dfrac{3}{5} = \dfrac{8}{5}\).
(i) \(\dfrac{4}{13}\)
(ii) \(\dfrac{5}{8}\)
| \(\dfrac{9}{4} \div \dfrac{10}{3}\) | \(=\) | \(\dfrac{9}{4} \times \dfrac{3}{10}\) |
| \(\) | \(=\) | \(\dfrac{27}{40}\) |
\(\dfrac{27}{40}\): less than \(1\), because \(2\dfrac{1}{4}\) is smaller than \(3\dfrac{1}{3}\).
| \(\dfrac{11}{2} \times \dfrac{4}{7}\) | \(=\) | \(\dfrac{11}{1} \times \dfrac{2}{7}\) |
| \(\) | \(=\) | \(\dfrac{22}{7} = 3\dfrac{1}{7}\) |
\(3\dfrac{1}{7}\)
| \(\dfrac{21}{5} \times \dfrac{4}{7}\) | \(=\) | \(\dfrac{3}{5} \times \dfrac{4}{1}\) |
| \(\) | \(=\) | \(\dfrac{12}{5} = 2\dfrac{2}{5}\) |
Estimate \(4 \div 2 = 2\). The answer \(2\dfrac{2}{5}\) is close.
(ii) A cyclist rides \(22\dfrac{1}{2}\) km in \(1\dfrac{1}{4}\) hours. What is her average speed?
| \(\dfrac{27}{4} \times \dfrac{8}{9}\) | \(=\) | \(6\) |
| \(\dfrac{45}{2} \times \dfrac{4}{5}\) | \(=\) | \(18\) |
(i) \(6\) shelves
(ii) \(18\) km/h
(i) How many whole pieces does she get?
(ii) How much more hose would she need for one more piece?
| \(\dfrac{11}{1} \times \dfrac{5}{8}\) | \(=\) | \(\dfrac{55}{8} = 6\dfrac{7}{8}\) |
| \(6 \times \dfrac{8}{5}\) | \(=\) | \(\dfrac{48}{5} = 9\dfrac{3}{5}\) |
| \(11 - 9\dfrac{3}{5}\) | \(=\) | \(1\dfrac{2}{5}\) |
(i) \(6\) pieces
(ii) \(1\dfrac{2}{5}\) m is left, so she needs \(1\dfrac{3}{5} - 1\dfrac{2}{5} = \dfrac{1}{5}\) m more.
Common pitfalls
Frequently asked questions
How do you divide mixed numerals?
Change both to improper fractions, then keep, change, flip, cancel and multiply.
What is the reciprocal of a mixed numeral?
Change it to an improper fraction, then flip. The reciprocal of \(3\dfrac{1}{4}\) is \(\dfrac{4}{13}\).
How do you estimate a division of mixed numerals?
Round each to the nearest whole number. \(4\dfrac{1}{5} \div 1\dfrac{3}{4} \approx 4 \div 2 = 2\).
Does dividing always make a number smaller?
No. Dividing by a number between \(0\) and \(1\) makes it bigger.
How do you find average speed?
Distance divided by time. \(22\dfrac{1}{2} \div 1\dfrac{1}{4} = 18\) km/h.