REAL NUMBERS AND RATIO

QUESTION 1 [6 marks] — Criterion B Medium

Investigate what happens to a ratio when both terms are multiplied by the same scale factor.

a. Starting with the ratio $3:5$, multiply both terms by 2, then by 3, then by 4. Record each resulting ratio.

[2]

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$\times2$: $6:10$. $\times3$: $9:15$. $\times4$: $12:20$.

b. Simplify each of your three new ratios back to simplest form. What do you notice?

[2]

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$6:10$ simplifies to $3:5$. $9:15$ simplifies to $3:5$. $12:20$ simplifies to $3:5$. Every scaled ratio simplifies back to the original ratio.

c. Explain why this happens, and state a general rule for ratio $a:b$ scaled by a factor $k$.

[2]

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Multiplying both terms of a ratio by the same factor doesn't change the underlying proportion between them — it's equivalent to multiplying a fraction's numerator and denominator by the same number, which doesn't change its value. General rule: $a:b$ is equivalent to $ka:kb$ for any $k>0$.

QUESTION 2 [5 marks] — Criterion C Medium

A garden bed is to have soil and compost mixed in the ratio $5:2$. The gardener has $35$ litres of soil available.

a. Show clear working to find how much compost is needed to maintain the correct ratio, and state the total volume of mix produced.

[3]

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Since soil corresponds to the '5' part of the ratio: $1$ part $= 35\div5=7$ litres. Compost needed (2 parts) $=2\times7=14$ litres. Total mix $=35+14=49$ litres.

b. Explain, using the ratio, how you would check that your answer maintains the exact $5:2$ proportion.

[2]

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Checking: $35:14$. Dividing both by their HCF (7): $35\div7=5$, $14\div7=2$, giving $5:2$ — confirming the ratio is maintained.

QUESTION 3 [6 marks] — Criterion D Medium

A map has a scale of $1:25000$, meaning 1 cm on the map represents 25000 cm (250 m) in real life. The distance between two towns on the map measures 8.4 cm.

a. Find the real-life distance between the two towns, giving your answer in kilometres.

[3]

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Real distance $= 8.4 \times 25000 = 210000$ cm $= 2100$ m $= 2.1$ km.

b. A hiker plans to walk between the towns at an average speed of 5 km/h, starting at 9:00 am, and wants to arrive before 10:00 am. Based on your answer to (a), will they arrive on time? Justify your answer.

[3]

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Time needed $= 2.1 \div 5 = 0.42$ hours $= 25.2$ minutes. Starting at 9:00 am, they would arrive at approximately 9:25 am, which is before 10:00 am — so yes, they will arrive on time, with about 35 minutes to spare.

Ratio