Answer Key
8.1 Fractions review
Pack A — Answers
| # | Question | Answer |
|---|---|---|
| 1 | Convert the improper fraction $\dfrac{17}{5}$ to a mixed numeral. | $3\tfrac{2}{5}$ |
| 2 | Convert the mixed numeral $4\tfrac{2}{3}$ to an improper fraction. | $\dfrac{14}{3}$ |
| 3 | Simplify the fraction $\dfrac{18}{24}$ to its simplest form. | $\dfrac{3}{4}$ |
| 4 | Write $\dfrac{3}{4}$ as an equivalent fraction with denominator 20. | $\dfrac{15}{20}$ |
| 5 | Calculate $\dfrac{2}{7} + \dfrac{3}{7}$ and simplify if possible. | $\dfrac{5}{7}$ |
| 6 | Calculate $\dfrac{1}{3} + \dfrac{1}{4}$. Give your answer in simplest form. | $\dfrac{7}{12}$ |
| 7 | Calculate $\dfrac{2}{3} \times \dfrac{3}{4}$. Give your answer in simplest form. | $\dfrac{1}{2}$ |
| 8 | Calculate $\dfrac{3}{4} \div \dfrac{1}{2}$. Give your answer in simplest form. | $\dfrac{3}{2}$ (or $1\tfrac{1}{2}$) |
| 9 | Write 35% as (a) a decimal and (b) a fraction in its simplest form. | (a) 0.35 (b) $\dfrac{7}{20}$ |
| 10 | Write $\dfrac{3}{4}$ as a decimal and as a percentage. | 0.75 and 75% |
| 11 | Calculate $\dfrac{5}{6} - \dfrac{1}{4}$. Give your answer in simplest form. | $\dfrac{7}{12}$ |
| 12 | Calculate $2\tfrac{1}{3} + 1\tfrac{1}{6}$. | $3\tfrac{1}{2}$ |
| 13 | Find $\dfrac{3}{5}$ of 600 g. | 360 g |
| 14 | Calculate $\dfrac{3}{8} \times 12$ where $n$ is a whole number. Give the answer in simplest form. | $\tfrac{9}{2}$ (or $4\tfrac{1}{2}$) |
| 15 | Calculate $\dfrac{4}{5} \div 8$. | $\dfrac{1}{10}$ |
| 16 | Order from smallest to largest: $\dfrac{2}{3}$, $\dfrac{5}{8}$, $\dfrac{7}{12}$. | $\dfrac{7}{12} < \dfrac{5}{8} < \dfrac{2}{3}$ |
| 17 | Convert the recurring decimal $0.\overline{3}$ to a fraction in its simplest form. | $\dfrac{1}{3}$ |
| 18 | Calculate $\dfrac{1}{2} + \dfrac{1}{3} - \dfrac{1}{6}$. | $\dfrac{2}{3}$ |
| 19 | Write 0.45 as a fraction in its simplest form. | $\dfrac{9}{20}$ |
| 20 | Calculate $2\tfrac{1}{4} \times \dfrac{4}{5}$. | $\dfrac{9}{5}$ (or $1\tfrac{4}{5}$) |
| 21 | A pizza is cut into 8 equal slices. Tomás eats $\tfrac{3}{8}$, and his sister eats $\tfrac{1}{4}$ of the whole pizza. What fraction of the pizza is left? | $\dfrac{3}{8}$ |
| 22 | A bottle holds 600 ml. Joëlle drinks $\tfrac{2}{5}$ of it on the way to school and $\tfrac{1}{4}$ of the remainder at break. How many ml are left? | 270 ml |
| 23 | Find $60\%$ of $\dfrac{3}{4}$ of 200 chf. | 90 chf |
| 24 | Calculate $\dfrac{2}{3} \div \dfrac{4}{9} + \dfrac{1}{2}$. | 2 |
| 25 | In a school survey, $\tfrac{2}{5}$ of the students chose Maths and 30% chose English. The rest chose Science. What fraction chose Science? Give your answer as a fraction in simplest form. | $\dfrac{3}{10}$ |
| 26 | Calculate $5\tfrac{1}{4} - 2\tfrac{1}{2}$. | $2\tfrac{3}{4}$ |
| 27 | Order from smallest to largest: 0.6, $\tfrac{5}{8}$, 62%. | $0.6 < 0.62 < \dfrac{5}{8}$ |
| 28 | A recipe for 8 pancakes uses $\tfrac{3}{4}$ cup of flour. How much flour is needed for 12 pancakes? Give the answer as a mixed numeral if possible. | $1\tfrac{1}{8}$ cups |
| 29 | Calculate $\left(\dfrac{1}{2} + \dfrac{1}{3}\right) \times \dfrac{6}{5}$. | 1 |
| 30 | Write 12.5% as a fraction in its simplest form. | $\dfrac{1}{8}$ |
| 31 | Calculate $\dfrac{\tfrac{2}{3} + \tfrac{1}{4}}{\tfrac{5}{6} - \tfrac{1}{3}}$. | $\dfrac{11}{6}$ (or $1\tfrac{5}{6}$) |
| 32 | A jug is $\tfrac{2}{3}$ full of juice. After pouring out 600 ml, it is $\tfrac{1}{4}$ full. How many ml does the jug hold when full? | 1440 ml |
| 33 | Convert the recurring decimal $0.\overline{27}$ to a fraction in its simplest form. | $\dfrac{3}{11}$ |
| 34 | A test has 60 questions. Léa answered $\tfrac{3}{5}$ of them correctly. Of the remainder, $\tfrac{1}{4}$ were left blank and the rest were wrong. How many were wrong? | 18 |
| 35 | Show that $0.\overline{9} = 1$ by an algebraic argument. | Let $x = 0.\overline{9}$. Then $10x = 9.\overline{9}$. Subtracting: $9x = 9$, so $x = 1$. |
| 36 | A pile of marbles is shared among three children. Anna takes $\tfrac{1}{4}$, Ben takes $\tfrac{2}{5}$ of what is left, and Carla takes the remaining 18. How many marbles were in the pile? | 40 |
| 37 | Two fractions $\tfrac{a}{b}$ and $\tfrac{c}{d}$ are equivalent if $ad = bc$. Determine whether $\tfrac{12}{18}$ and $\tfrac{20}{30}$ are equivalent, and explain. | Yes — equivalent (cross product $360 = 360$). |
| 38 | Find a fraction $\tfrac{p}{q}$ between $\tfrac{3}{8}$ and $\tfrac{1}{2}$ with a denominator less than 10. | $\dfrac{2}{5}$ (or $\dfrac{3}{7}$) |
| 39 | A fraction $\tfrac{n}{n+5}$ simplifies to $\tfrac{3}{4}$. Find $n$. | $n = 15$ |
| 40 | Show that the sum $1 + \tfrac{1}{2} + \tfrac{1}{4} + \tfrac{1}{8}$ can be written as a single improper fraction, and explain whether the pattern $1 + \tfrac{1}{2} + \tfrac{1}{4} + \ldots + \tfrac{1}{2^n}$ ever reaches 2. | Sum $= \dfrac{15}{8}$; the pattern approaches 2 but never reaches it (it gets infinitesimally close). |
Pack B — Answers
| # | Question | Answer |
|---|---|---|
| 1 | Convert the improper fraction $\dfrac{23}{4}$ to a mixed numeral. | $5\tfrac{3}{4}$ |
| 2 | Convert the mixed numeral $3\tfrac{5}{6}$ to an improper fraction. | $\dfrac{23}{6}$ |
| 3 | Simplify the fraction $\dfrac{28}{42}$ to its simplest form. | $\dfrac{2}{3}$ |
| 4 | Write $\dfrac{2}{5}$ as an equivalent fraction with denominator 30. | $\dfrac{12}{30}$ |
| 5 | Calculate $\dfrac{3}{12} + \dfrac{5}{12}$ and simplify if possible. | $\dfrac{8}{12} = \dfrac{2}{3}$ |
| 6 | Calculate $\dfrac{2}{5} + \dfrac{1}{2}$. Give your answer in simplest form. | $\dfrac{9}{10}$ |
| 7 | Calculate $\dfrac{4}{5} \times \dfrac{5}{6}$. Give your answer in simplest form. | $\dfrac{2}{3}$ |
| 8 | Calculate $\dfrac{5}{6} \div \dfrac{5}{12}$. Give your answer in simplest form. | 2 |
| 9 | Write 65% as (a) a decimal and (b) a fraction in its simplest form. | (a) 0.65 (b) $\dfrac{13}{20}$ |
| 10 | Write $\dfrac{7}{20}$ as a decimal and as a percentage. | 0.35 and 35% |
| 11 | Calculate $\dfrac{7}{10} - \dfrac{2}{5}$. Give your answer in simplest form. | $\dfrac{3}{10}$ |
| 12 | Calculate $3\tfrac{1}{4} + 2\tfrac{1}{2}$. | $5\tfrac{3}{4}$ |
| 13 | Find $\dfrac{2}{3}$ of 450 g. | 300 g |
| 14 | Calculate $\dfrac{5}{9} \times 27$ where $n$ is a whole number. Give the answer in simplest form. | 15 |
| 15 | Calculate $\dfrac{9}{10} \div 6$. | $\dfrac{3}{20}$ |
| 16 | Order from smallest to largest: $\dfrac{2}{3}$, $\dfrac{5}{8}$, $\dfrac{7}{12}$. | $\dfrac{7}{12} < \dfrac{5}{8} < \dfrac{2}{3}$ |
| 17 | Convert the recurring decimal $0.\overline{6}$ to a fraction in its simplest form. | $\dfrac{2}{3}$ |
| 18 | Calculate $\dfrac{3}{4} + \dfrac{1}{6} - \dfrac{1}{12}$. | $\dfrac{5}{6}$ |
| 19 | Write 0.625 as a fraction in its simplest form. | $\dfrac{5}{8}$ |
| 20 | Calculate $3\tfrac{1}{3} \times \dfrac{3}{4}$. | $\dfrac{5}{2}$ (or $2\tfrac{1}{2}$) |
| 21 | A pizza is cut into 8 equal slices. Tomás eats $\tfrac{3}{8}$, and his sister eats $\tfrac{1}{4}$ of the whole pizza. What fraction of the pizza is left? | $\dfrac{3}{8}$ |
| 22 | A bottle holds 800 ml. Joëlle drinks $\tfrac{2}{5}$ of it on the way to school and $\tfrac{1}{4}$ of the remainder at break. How many ml are left? | 360 ml |
| 23 | Find $25\%$ of $\dfrac{2}{5}$ of 400 chf. | 40 chf |
| 24 | Calculate $\dfrac{3}{5} \div \dfrac{6}{25} + \dfrac{1}{4}$. | $\dfrac{11}{4}$ (or $2\tfrac{3}{4}$) |
| 25 | In a school survey, $\tfrac{2}{5}$ of the students chose Maths and 30% chose English. The rest chose Science. What fraction chose Science? Give your answer as a fraction in simplest form. | $\dfrac{67}{300}$ |
| 26 | Calculate $6\tfrac{1}{3} - 3\tfrac{5}{6}$. | $2\tfrac{1}{2}$ |
| 27 | Order from smallest to largest: 0.6, $\tfrac{5}{8}$, 62%. | $\dfrac{4}{9} < 0.45 < 0.47$ |
| 28 | A recipe for 8 pancakes uses $\tfrac{3}{4}$ cup of flour. How much flour is needed for 12 pancakes? Give the answer as a mixed numeral if possible. | $1\tfrac{1}{9}$ cups |
| 29 | Calculate $\left(\dfrac{1}{4} + \dfrac{1}{6}\right) \times \dfrac{12}{5}$. | 1 |
| 30 | Write 87.5% as a fraction in its simplest form. | $\dfrac{7}{8}$ |
| 31 | Calculate $\dfrac{\tfrac{2}{3} + \tfrac{1}{4}}{\tfrac{5}{6} - \tfrac{1}{3}}$. | $\dfrac{11}{6}$ |
| 32 | A jug is $\tfrac{2}{3}$ full of juice. After pouring out 600 ml, it is $\tfrac{1}{4}$ full. How many ml does the jug hold when full? | 1200 ml |
| 33 | Convert the recurring decimal $0.\overline{45}$ to a fraction in its simplest form. | $\dfrac{5}{11}$ |
| 34 | A test has 60 questions. Léa answered $\tfrac{3}{5}$ of them correctly. Of the remainder, $\tfrac{1}{4}$ were left blank and the rest were wrong. How many were wrong? | 25 |
| 35 | Show that $0.\overline{9} = 1$ by an algebraic argument. | See Pack A. |
| 36 | A pile of marbles is shared among three children. Anna takes $\tfrac{1}{4}$, Ben takes $\tfrac{2}{5}$ of what is left, and Carla takes the remaining 18. How many marbles were in the pile? | 36 |
| 37 | Two fractions $\tfrac{a}{b}$ and $\tfrac{c}{d}$ are equivalent if $ad = bc$. Determine whether $\tfrac{12}{18}$ and $\tfrac{20}{30}$ are equivalent, and explain. | Yes — equivalent (cross product $225 = 225$). |
| 38 | Find a fraction $\tfrac{p}{q}$ between $\tfrac{3}{8}$ and $\tfrac{1}{2}$ with a denominator less than 10. | $\dfrac{2}{7}$ (any value in the range with denominator < 10) |
| 39 | A fraction $\tfrac{n}{n+5}$ simplifies to $\tfrac{3}{4}$. Find $n$. | $n = 14$ |
| 40 | Show that the sum $1 + \tfrac{1}{2} + \tfrac{1}{4} + \tfrac{1}{8}$ can be written as a single improper fraction, and explain whether the pattern $1 + \tfrac{1}{2} + \tfrac{1}{4} + \ldots + \tfrac{1}{2^n}$ ever reaches 2. | See Pack A. |
Problems — Worked Solutions
**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.
(a) $\tfrac{1}{6}$ each (b) $\tfrac{3}{12}, \tfrac{2}{12}, \tfrac{2}{12}, \tfrac{2}{12}, \tfrac{1}{12}, \tfrac{2}{12}$ (Fati gets the last 2) (c) See working
**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?
(a) $1\tfrac{1}{8}, \tfrac{3}{4}, \tfrac{3}{8}$ (b) $\tfrac{1}{2}, \tfrac{1}{3}, \tfrac{1}{6}$ (c) 24 pancakes
**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 $0.\overline{9} = 1$.
(a) $\tfrac{4}{9}$, $\tfrac{3}{11}$, $\tfrac{16}{999}$ (b) (i) $\tfrac{2}{3}$; (ii) correct; (iii) wrong — should be $\tfrac{21}{999}$ (or $\tfrac{7}{333}$) (c) See working
**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) $1 - \tfrac{1}{2}\left(\tfrac{3}{4} + \tfrac{1}{6}\right)$
(a) $\tfrac{4}{3}$ (b) $\tfrac{3}{5}$ (c) $\tfrac{13}{24}$
**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?
(a) $\tfrac{1}{2}$ (b) $\tfrac{3}{8}$ (c) 562.5 ml
**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?
(a) 216 (b) 162 (c) $\tfrac{9}{20}$
**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 $x$.
(a) Yes (b) $x = 4$ (c) $x = 2$
**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.
(a) 260 EUR (b) 144 chf (c) $\tfrac{12}{13}$ chf ≈ 0.9231 chf
**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.
(a) $\tfrac{1}{6}$ (b) $\tfrac{1}{4}$ (c) $\tfrac{5}{12}$; 2 h 24 min
**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 $\tfrac{1}{2}$?
(a) $\tfrac{3}{2}, \tfrac{5}{6}, \tfrac{7}{12}, \tfrac{9}{20}, \tfrac{11}{30}$ (b) See working (c) For $n \geq 4$
**Fraction–decimal–percentage triangle.** Complete the following table. | Fraction | Decimal | Percentage | |----------|---------|------------| | $\tfrac{3}{8}$ | ? | ? | | ? | 0.65 | ? | | ? | ? | 12% | | ? | 1.2 | ? | Give fractions in simplest form.
Row 1: $\tfrac{3}{8}$, 0.375, 37.5%. Row 2: $\tfrac{13}{20}$, 0.65, 65%. Row 3: $\tfrac{3}{25}$, 0.12, 12%. Row 4: $\tfrac{6}{5}$ (or $1\tfrac{1}{5}$), 1.2, 120%.
**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.
(a) 2.5 turns (b) $\tfrac{34}{32} = \tfrac{17}{16}$ (≈ 1.06) (c) $\tfrac{50}{11}$ (≈ 4.55)