7.5 Fractions & Percentages

**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.

**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?

**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?

**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 $\dfrac{3}{7}$.

**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.

**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?

**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.

**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.

**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.

**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.

**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?

**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?