Problem-solving Pack

Mathematics

Year 7 · 7.5 Fractions & Percentages

Problem-solving Pack

Name: _________________________________

Date: _________________ Class: ___________

These problems are designed to challenge you. Read each question carefully. Show all your reasoning — a correct answer without working receives no credit.

1Problem 1 of 12

Compound percentage. A jacket originally costs £80. It is reduced by 25% in a sale. The following week, the sale price is reduced by a further 10%.

What is the final price? Is the total reduction 35%? Explain why or why not.

Working space

2Problem 2 of 12

Fraction wall. In a school of 360 students:
-

\dfrac{3}{8}

study French.
-

\dfrac{1}{4}

of those French students also study Spanish.
- The remaining French students study French only.

How many students study French only?

Working space

3Problem 3 of 12

Pipe filling. Pipe A fills a tank in 4 hours. Pipe B fills the same tank in 6 hours.

If both pipes are open together, how long does it take to fill the tank?

Working space

4Problem 4 of 12

Fractions that sum to 1. The ancient Egyptians only wrote fractions with numerator 1 (called "unit fractions"), such as

\frac{1}{2}

\frac{1}{3}

\frac{1}{4}

.

(a) Write

\dfrac{5}{6}

as a sum of two different unit fractions.
(b) Write

\dfrac{7}{12}

as a sum of two different unit fractions.
(c) Show that

\dfrac{2}{n} = \dfrac{1}{n} + \dfrac{1}{n}

does not count (same fraction twice). Instead, find unit fractions

\dfrac{1}{a}

and

\dfrac{1}{b}

with

a < b

such that

\dfrac{1}{a} + \dfrac{1}{b} = \dfrac{2}{9}

.
(d) Is it always possible to write any proper fraction as a sum of unit fractions? Try

\dfrac{3}{7}

Working space

5Problem 5 of 12

Percentage chain. A shop increases all prices by 20% in January. In July, it reduces all prices by 20%.

A customer claims "the prices are back to where they started." Is she correct? Give a numerical example to support your answer.

Working space

6Problem 6 of 12

Fraction of the way. A road is 24 km long. A cyclist has completed

\dfrac{5}{8}

of the journey.

(a) How far has she cycled?
(b) What fraction of the journey remains?
(c) If the remaining distance takes 45 minutes, what is her average speed in km/h for that section?

Working space

7Problem 7 of 12

Average with constraints. Five different positive integers have mean 6 and median 5.

(a) What is their sum?
(b) The smallest is 1. Find all possible sets of five integers satisfying every condition.
(c) If the smallest must be 2 instead of 1, does a valid set still exist? Explain.

Working space

8Problem 8 of 12

Simplifying with prime factors.

(a) Write 126 and 210 each as a product of prime factors.
(b) Hence find the HCF of 126 and 210.
(c) Use the HCF to simplify

\dfrac{126}{210}

fully.

Working space

9Problem 9 of 12

Fractions and algebra. The perimeter of a rectangle is 15 cm. One side has length

2\dfrac{1}{4}

cm.

Find the length of the other side as a mixed number.

Working space

10Problem 10 of 12

Farey sequence investigation. A Farey sequence

F_n

contains all fractions between 0 and 1 (inclusive) with denominators at most

n

, written in ascending order.

F_3

\dfrac{0}{1},\ \dfrac{1}{3},\ \dfrac{1}{2},\ \dfrac{2}{3},\ \dfrac{1}{1}

(a) Write out

F_4

in full (all fractions with denominators 1, 2, 3, or 4, in order).
(b) Pick any two adjacent fractions in

F_4

, say

\dfrac{a}{b}

and

\dfrac{c}{d}

. Calculate

bc - ad

. What do you notice?
(c) For two adjacent Farey fractions

\dfrac{a}{b}

and

\dfrac{c}{d}

, the mediant is

\dfrac{a+c}{b+d}

. Find the mediant of

\dfrac{1}{3}

and

\dfrac{1}{2}

. Is it between them?
(d) Verify that the mediant of two adjacent Farey fractions always lies strictly between them.

Working space

11Problem 11 of 12

Mixed units. A recipe needs

\dfrac{3}{4}

kg of flour for 12 biscuits.

(a) How much flour is needed for 20 biscuits?
(b) A bag holds 1.5 kg. What fraction of the bag is used for 20 biscuits?
(c) What percentage of the bag is left over?

Working space

12Problem 12 of 12

Jamie's monthly budget. Jamie earns £960 per month.

- He saves

\dfrac{1}{4}

of his earnings.
- He spends 35% of the remainder on rent.
- He spends

\dfrac{2}{5}

of what is left on food and bills.

How much does Jamie have left each month for discretionary spending?

Working space