These problems combine ideas – you will need to plan your own steps, and sometimes form your own equation. There is a worked example to show you how to start; then it is over to you. Stuck? Hints are at the foot of the sheet. No calculator.
Worked example: how to start
In a class, $\dfrac{2}{3}$ of the students walk to school. Of those who walk, $\dfrac{1}{4}$ cycle the last part of the way. What fraction of the whole class cycle the last part of the way?
“$\dfrac{1}{4}$ of those who walk” means $\dfrac{1}{4}$ of $\dfrac{2}{3}$, so multiply: $\dfrac{1}{4}\times\dfrac{2}{3}$.
1A jug holds $\dfrac{3}{4}$ of a litre of juice. Rana pours out $\dfrac{1}{3}$ of the juice. How much juice, in litres, is left in the jug?[3 marks]
Answer:
2Three quarters of a number is $18$. Work out one third of the same number.[3 marks]
Answer:
3A recipe needs $\dfrac{2}{3}$ of a cup of flour for one cake. Priya has $5$ cups of flour. What is the greatest number of whole cakes she can make, and how much flour is left over?[4 marks]
Answer:
4Adam spends $\dfrac{1}{4}$ of his money on a book, then $\dfrac{2}{5}$ of the money he has left on a pen. He now has £$9$ left. How much money did Adam have to start with?[4 marks]
Answer:
5In a bag of counters, $\dfrac{2}{5}$ are red. When $6$ more red counters are added, exactly half of the counters are red. How many counters were in the bag to start with?[5 marks]
Answer:
★Extension: A fraction is equivalent to $\dfrac{2}{5}$. When $4$ is added to both its numerator and its denominator, the new fraction is equal to $\dfrac{1}{2}$. Work out the original fraction.
Stuck? Hints (don't peek unless you need to)1. Either find the fraction of a litre poured out first, or find the fraction of the juice that remains and take that fraction of $\dfrac{3}{4}$.2. Find the whole number first: if $\dfrac{3}{4}$ of it is 18, what is $\dfrac{1}{4}$ of it?3. Divide $5$ by $\dfrac{2}{3}$ to see how many cakes’ worth of flour there is; the whole-number part is the number of cakes.4. Work out the single fraction of his ORIGINAL money that is left after both purchases, then use that it equals £9.5. Call the starting number $n$. Write the number of red counters before, and after adding 6, in terms of $n$, and use that the new fraction is $\dfrac{1}{2}$.
Solutions & mark scheme · Fractions: Four Operations
Total: 19 marks
Award the marks shown for each correct step; many of these have more than one valid route, so give method marks for any correct working.
1A jug holds $\dfrac{3}{4}$ of a litre of juice. Rana pours out $\dfrac{1}{3}$ of the juice. How much juice, in litres, is left in the jug?[3]
Model solution
Juice poured out $=\dfrac{1}{3}\times\dfrac{3}{4}=\dfrac{3}{12}=\dfrac{1}{4}$ litre.
Juice left $=\dfrac{3}{4}-\dfrac{1}{4}=\dfrac{2}{4}=\dfrac{1}{2}$ litre.
Answer: $\dfrac{1}{2}$ litre
Marks
✔1Poured out $=\frac13\times\frac34=\frac14$ (or remaining fraction $\frac23$)
✔1Subtracts from $\frac34$
✔1$\frac12$ litre
2Three quarters of a number is $18$. Work out one third of the same number.[3]
Model solution
$\dfrac{1}{4}$ of the number $=18\div 3=6$, so the number $=6\times 4=24$.
$\dfrac{1}{3}$ of $24=24\div 3=8$.
Answer: 8
Marks
✔1Finds $\frac14$ of the number $=6$
✔1Whole number $=24$
✔1$8$
3A recipe needs $\dfrac{2}{3}$ of a cup of flour for one cake. Priya has $5$ cups of flour. What is the greatest number of whole cakes she can make, and how much flour is left over?[4]
Model solution
$5\div\dfrac{2}{3}=5\times\dfrac{3}{2}=\dfrac{15}{2}=7\dfrac{1}{2}$, so she can make $7$ whole cakes.
Flour used $=7\times\dfrac{2}{3}=\dfrac{14}{3}=4\dfrac{2}{3}$ cups.
Flour left $=5-\dfrac{14}{3}=\dfrac{15}{3}-\dfrac{14}{3}=\dfrac{1}{3}$ of a cup.
Answer: 7 cakes, $\dfrac{1}{3}$ cup left
Marks
✔1$5\div\frac23=\frac{15}{2}$
✔1$7$ whole cakes
✔1Flour used $=\frac{14}{3}$
✔1$\frac13$ cup left
4Adam spends $\dfrac{1}{4}$ of his money on a book, then $\dfrac{2}{5}$ of the money he has left on a pen. He now has £$9$ left. How much money did Adam have to start with?[4]
Model solution
After the book, $\dfrac{3}{4}$ of the money is left.
He spends $\dfrac{2}{5}$ of that, so he keeps $\dfrac{3}{5}$ of it: fraction left $=\dfrac{3}{5}\times\dfrac{3}{4}=\dfrac{9}{20}$.
$\dfrac{9}{20}$ of the total $=9$, so the total $=9\div\dfrac{9}{20}=9\times\dfrac{20}{9}=20$. He had £$20$.
Answer: £20
Marks
✔1$\frac34$ left after the book
✔1Fraction left $=\frac35\times\frac34=\frac{9}{20}$
✔1$\frac{9}{20}$ of total $=9$
✔1£$20$
5In a bag of counters, $\dfrac{2}{5}$ are red. When $6$ more red counters are added, exactly half of the counters are red. How many counters were in the bag to start with?[5]
Model solution
Let the starting number be $n$. Red counters at the start $=\dfrac{2}{5}n$.
After adding $6$ red: $\dfrac{\frac{2}{5}n+6}{n+6}=\dfrac{1}{2}$.
Cross-multiply: $2\left(\dfrac{2}{5}n+6\right)=n+6$, so $\dfrac{4}{5}n+12=n+6$.
$12-6=n-\dfrac{4}{5}n$, so $6=\dfrac{1}{5}n$, giving $n=30$.
Answer: 30 counters
Marks
✔1Red at start $=\frac25 n$
✔1Equation $\frac{\frac25 n+6}{n+6}=\frac12$
✔1Rearranges to $\frac45 n+12=n+6$
✔1$\frac15 n=6$
✔1$n=30$
★Extension
Write the original as $\dfrac{2k}{5k}$ for some whole number $k$.
$\dfrac{2k+4}{5k+4}=\dfrac{1}{2}$, so $2(2k+4)=5k+4$.
$4k+8=5k+4$, so $k=4$.
Original fraction $=\dfrac{2\times 4}{5\times 4}=\dfrac{8}{20}$.
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