S1 Mathematics Practice Book · Unit 4: Ratio and proportions

Unit 4 Ratio and proportions

Key unit competence: Solve problems involving ratios and proportions.

Key ideas

A ratio $a : b$ compares quantities of the same kind (in the same units). It can be written as a fraction $\frac{a}{b}$ and simplified by dividing by the highest common factor. Ratios have no units.
To share a quantity $Q$ in the ratio $a : b$ , split it into $a + b$ equal parts: the shares are $\frac{a}{a + b} Q$ and $\frac{b}{a + b} Q$ .
A scale $1 : n$ means 1 unit on the drawing represents $n$ units in reality.
A proportion $a : b = c : d$ states that two ratios are equal. $b$ and $c$ are the means, $a$ and $d$ the extremes; the product of the means equals the product of the extremes: $a d = b c$ .
Direct proportion: both quantities increase together in the same ratio; $y = k x$ ; the graph is a straight line through the origin.
Inverse proportion: when one doubles, the other halves; $x y = k$ , so $y = \frac{k}{x}$ ; the graph is a curve (hyperbola).
Mixtures: average price $= \frac{total cost}{total quantity}$ . To find unknown quantities, form the cost equation $P_{1} x + P_{2} (T - x) = A T$ .

Formulas to know

\frac{a}{b} = \frac{c}{d} ⟺ a d = b c

share = \frac{a}{a + b} \times Q

y = k x (direct), x y = k (inverse)

average price = \frac{total cost}{total quantity}

Common misconceptions

"The ratio 2 m : 50 cm is 2 : 50." No: change to the same units first: 200 cm : 50 cm = 4 : 1.
"More workers take more days." No: workers and days for the same job are in inverse proportion.
"The average price of a mixture is the average of the two prices." Only when equal quantities are mixed; otherwise divide total cost by total quantity.

Worked examples

Example 1. Share 38 400 FRW between Linda and Jean in the ratio 5 : 7.

Solution

Total parts $5 + 7 = 12$ . Linda gets $\frac{5}{12} \times 38400 = 16000$ FRW, Jean gets $\frac{7}{12} \times 38400 = 22400$ FRW.

Example 2. A trader mixes rice at 600 FRW/kg and rice at 900 FRW/kg to get 20 kg at 750 FRW/kg. How much of each?

Solution

Let $x$ kg at 600. Then $600 x + 900 (20 - x) = 750 \times 20 = 15000$ .

$18000 - 300 x = 15000$ , $x = 10$ . So 10 kg of each type.

Practice questions

Easier and harder questions are mixed, just like in a real exam.

A business lady mixes 15 kg of sorghum flour with 9 kg of cassava flour. Write the ratio in simplest form.
Show answer
$15 : 9 = 5 : 3$ .
Common mistake: Writing $9 : 15$ (wrong order).
Coffee at 3 000 FRW/kg and 4 500 FRW/kg is mixed to make 30 kg worth 3 500 FRW/kg. Find both quantities.
Show answer
$3000 x + 4500 (30 - x) = 105000$ ; $135000 - 1500 x = 105000$ ; $x = 20$ . 20 kg at 3 000 and 10 kg at 4 500.
Common mistake: Forgetting to multiply the average price by the total quantity.
Share 72 000 FRW among three children in the ratio $2 : 3 : 7$ .
Show answer
12 parts; one part = 6 000 FRW. Shares: 12 000, 18 000 and 42 000 FRW.
Common mistake: Dividing by 3 (number of children) instead of 12 (number of parts).
A trader mixes 10 kg of tea at 2 400 FRW/kg with 15 kg at 2 000 FRW/kg. At what price per kg should she sell the mixture to break even?
Show answer
Cost $24000 + 30000 = 54000$ FRW for 25 kg: 2 160 FRW/kg.
Common mistake: Averaging the two prices to get 2 200 FRW/kg.
The distance between two towns on a map of scale $1 : 200000$ is 6.5 cm. Find the real distance in km.
Show answer
$6.5 \times 200000 = 1300000$ cm $= 13$ km.
Common mistake: Converting cm to km incorrectly (1 km = 100 000 cm).
A table of time (h) and distance (km) at constant speed: (1, 60), (2, 120), (3, 180). Is this a direct proportion? Give the constant.
Show answer
Yes: $\frac{d}{t} = 60$ every time, so $d = 60 t$ ; $k = 60$ km/h.
Common mistake: Checking only one pair of values.
Use the cross product to find $x$ if $\frac{5}{x} = \frac{15}{12}$ .
Show answer
$15 x = 60$ , so $x = 4$ .
Common mistake: Multiplying across the same row instead of diagonally.
12 men can dig a trench in 9 days. How many men are needed to dig it in 6 days?
Show answer
$12 \times 9 = 108$ man-days; $108 \div 6 = 18$ men.
Common mistake: Treating it as direct proportion and getting 8 men.

Matching and drag-and-drop

Matching

Match each item in column A with the correct answer in column B. Write the letter next to each number.

Column A	Column B
1. The ratio of boys to girls is $3 : 4$ and there are 21 boys. How many girls?	a. 400 FRW
2. 3 tomatoes cost 100 FRW. How much do 12 tomatoes cost?	b. 28
3. 72 000 FRW is shared among A, B and C in the ratio $2 : 3 : 4$ . How much does B get?	c. 24 000 FRW
4. 4 kg at 900 FRW/kg is mixed with 6 kg at 1 100 FRW/kg. Find the average price.	d. 1 020 FRW/kg

Show answer

1 → b (28) · 2 → a (400 FRW) · 3 → c (24 000 FRW) · 4 → d (1 020 FRW/kg)

One part $= 21 \div 3 = 7$ ; girls $= 4 \times 7 = 28$ .
12 is 4 times 3, so the cost is $4 \times 100 = 400$ FRW.
Step 1. Total parts: $2 + 3 + 4 = 9$ ; one part $= 72000 \div 9 = 8000$ .
Step 2. B gets $3 \times 8000 = 24000$ FRW.
Total cost $3600 + 6600 = 10200$ FRW for 10 kg: 1 020 FRW/kg.

Drag and drop
Choose the correct word or phrase from the box for each gap. One word in the box is not needed.
28 litres · 2 km · 15 days · 20 km
1. On a map of scale $1 : 50000$ , 4 cm represents .
2. 6 workers build a wall in 10 days. How many days would 4 workers take at the same rate? →
3. A car uses 8 litres of fuel per 100 km. How much fuel does it need for 350 km? →
Show answer
1: 2 km · 2: 15 days · 3: 28 litres
- $4 \times 50000 = 200000$ cm $= 2$ km.
- Step 1. Work $= 6 \times 10 = 60$ worker-days.
  Step 2. $60 \div 4 = 15$ days.
- Step 1. Fuel per km: $\frac{8}{100} = 0.08$ L.
  Step 2. $350 \times 0.08 = 28$ L.

Matching

Match each item in column A with the correct answer in column B. Write the letter next to each number.

Column A	Column B
1. If $y$ is inversely proportional to $x$ and $y = 6$ when $x = 4$ , find $k$ in $x y = k$ .	a. 1 728 FRW
2. Express $2 m : 50 cm$ in simplest form.	b. $4 : 1$
3. 5 kg of sugar costs 2 880 FRW. Find the cost of 3 kg.	c. 24
4. Solve $\frac{x}{8} = \frac{3}{4}$ .	d. $x = 6$

Show answer

1 → c (24) · 2 → b ( $4 : 1$ ) · 3 → a (1 728 FRW) · 4 → d ( $x = 6$ )

$k = x y = 4 \times 6 = 24$ .
Step 1. $2 m = 200$ cm.
Step 2. $200 : 50 = 4 : 1$ .
Step 1. 1 kg costs $2880 \div 5 = 576$ FRW.
Step 2. $3 \times 576 = 1728$ FRW.
Step 1. $4 x = 24$ .
Step 2. So $x = 6$ .

Drag and drop
Choose the correct word or phrase from the box for each gap. One word in the box is not needed.
200 FRW · 1 200 FRW/kg · 400 FRW · 8 days
1. Share 600 FRW in the ratio $1 : 2$ . What is the larger share? →
2. 12 workers build a wall in 10 days. How many days would 15 workers take at the same rate? →
3. 5 kg of beans at 800 FRW/kg are mixed with 3 kg of another type. The mixture costs 950 FRW/kg. Find the price of the second type. →
Show answer
1: 400 FRW · 2: 8 days · 3: 1 200 FRW/kg
- $\frac{2}{3} \times 600 = 400$ .
- Step 1. Total work: $12 \times 10 = 120$ worker-days (inverse proportion).
  Step 2. $120 \div 15 = 8$ days.
- Step 1. Total cost $8 \times 950 = 7600$ ; first type $4000$ .
  Step 2. $3 P = 3600$ .
  Step 3. $P = 1200$ .

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