8.1 Fractions review

**Pizza split.** A pizza is cut into 12 equal slices. Six friends share it. (a) What fraction of the pizza does each friend get if they share equally? (b) Alex gets 3 slices, Brigit 2, Chen 2, Dany 2, Eli 1. What fraction does each get? Sum the fractions to check they total 1. (c) Convert each fraction to a percentage.

**Recipe scaling with fractions.** A pancake recipe for 6 pancakes uses $\tfrac{3}{4}$ cup flour, $\tfrac{1}{2}$ cup milk and $\tfrac{1}{4}$ cup sugar. (a) How much of each ingredient is needed for 9 pancakes? (b) How much of each ingredient is needed for 4 pancakes? Express each answer as a fraction in simplest form. (c) A baker has 3 cups of flour and unlimited milk and sugar. How many pancakes can she make at most?

**Recurring decimals to fractions.** Recurring decimals can be converted using the "method of 9s": $0.\overline{a} = \tfrac{a}{9}$, $0.\overline{ab} = \tfrac{ab}{99}$, $0.\overline{abc} = \tfrac{abc}{999}$. (a) Convert $0.\overline{4}$, $0.\overline{27}$ and $0.\overline{016}$ to fractions in simplest form. (b) Which of these conversions are correct? For each incorrect one, give the correct fraction. (i) $0.\overline{6} = \tfrac{6}{9}$ (ii) $0.\overline{37} = \tfrac{37}{99}$ (iii) $0.\overline{021} = \tfrac{21}{99}$ (c) Show algebraically that $0.\overline{9} = 1$.

**Compound fractions.** Simplify each expression. Give the final answer as a fraction in its simplest form. (a) $\tfrac{2}{3} + \tfrac{1}{4} \times \tfrac{8}{3}$ (b) $\dfrac{\tfrac{5}{6} - \tfrac{1}{3}}{\tfrac{2}{3} + \tfrac{1}{6}}$ (c) $1 - \tfrac{1}{2}\left(\tfrac{3}{4} + \tfrac{1}{6}\right)$

**The remaining juice.** A bottle of juice is $\tfrac{5}{6}$ full. Alice drinks $\tfrac{1}{3}$ of the **bottle's** capacity. Then Ben drinks $\tfrac{1}{4}$ of what is left. (a) What fraction of the bottle's capacity is left after Alice drinks? (b) What fraction is left after Ben drinks? (c) If the bottle holds 1500 ml, how many ml are left?

**Fractions of fractions.** A school has 360 pupils. $\tfrac{2}{5}$ are in Lower School and the rest are in Upper School. $\tfrac{3}{4}$ of Upper School pupils study a second language. (a) How many pupils are in Upper School? (b) How many of those study a second language? (c) What fraction of the whole school studies a second language in Upper School?

**Cross-multiplication for equivalence.** Two fractions are equivalent if $\tfrac{a}{b} = \tfrac{c}{d}$ ⇔ $ad = bc$. (a) Test whether $\tfrac{9}{12}$ and $\tfrac{15}{20}$ are equivalent. (b) Find $x$ such that $\tfrac{x}{18} = \tfrac{2}{9}$. (c) Solve $\tfrac{x}{x+3} = \tfrac{2}{5}$ for $x$.

**Currency conversion.** 1 chf $= \tfrac{13}{12}$ EUR. (a) Convert 240 chf to euros. (b) Convert 156 EUR back to chf. (c) Convert 1 EUR to chf as a fraction in simplest form, then as a decimal to 4 d.p.

**Painting a fence.** Alex can paint a fence in 6 hours; Brigit can paint the same fence in 4 hours. (a) What fraction of the fence does Alex paint in 1 hour? (b) What fraction does Brigit paint in 1 hour? (c) If they work together, what fraction do they paint in 1 hour, and how long does the whole fence take? Give the answer in hours and minutes.

**Investigating $\tfrac{1}{n} + \tfrac{1}{n+1}$.** Look at the sum $\tfrac{1}{n} + \tfrac{1}{n+1}$ for various $n$. (a) Calculate the sum for $n = 1, 2, 3, 4, 5$. (b) Show algebraically that $\tfrac{1}{n} + \tfrac{1}{n+1} = \tfrac{2n + 1}{n(n+1)}$. (c) For which $n$ is the sum less than $\tfrac{1}{2}$?

**Fraction–decimal–percentage triangle.** Complete the following table. | Fraction | Decimal | Percentage | |----------|---------|------------| | $\tfrac{3}{8}$ | ? | ? | | ? | 0.65 | ? | | ? | ? | 12% | | ? | 1.2 | ? | Give fractions in simplest form.

**Bicycle gears.** A road bicycle has two front chainrings (34 teeth and 50 teeth) and a cassette of 11 sprockets (11 to 32 teeth). One full pedal turn means the chain advances by the number of teeth on the chainring; the rear wheel turns by (chainring ÷ sprocket) revolutions. (a) On a 50-tooth chainring and a 20-tooth sprocket, how many turns of the wheel per pedal turn? (b) What is the **smallest** gear ratio (lowest wheel-turns-per-pedal) the bike can achieve? (c) What is the **largest** gear ratio? Express each gear ratio as a fraction in simplest form.