Mastering Ratios: Practice Questions and Answers

Getting Started with Ratios

Before diving into exam questions, you'll want your basic tools ready: pencil, pen, ruler, protractor, compasses and eraser. The key to success with ratio problems is reading carefully and showing all your working - even if you make a mistake, you can still pick up marks for your method.

Don't get stuck on tricky questions for too long. Move on and come back later if you have time. Every question is worth attempting, and your workings often matter more than just the final answer.

Top Tip: Always check if your final ratios make sense - do the parts add up to the whole?

Simplifying Ratios and Basic Sharing

Simplifying ratios works just like simplifying fractions - find the highest common factor and divide both sides by it. For 25:35, both numbers divide by 5 to give 5:7. For 18:45, divide by 9 to get 2:5.

When dividing amounts in ratios, first add up all the ratio parts. To split £700 in the ratio 5:3:2, add the parts $5+3+2=10$, then divide the total by this sum $£700÷10=£70$. This gives you the value of one part.

Multiply each ratio number by this value: £70×5=£350, £70×3=£210, £70×2=£140. Quick check: do these add back to £700? Yes!

Remember: The ratio parts must add up to give you the original total amount.

Sharing Objects and Finding Quantities

Real-world ratio sharing problems follow the same pattern whether you're dividing sweets or pencils. Alex and Thomas share 30 sweets in the ratio 3:2 - add the parts $3+2=5$, divide the total $30÷5=6$, then multiply (Thomas gets 6×2=12 sweets).

For Sophie's pencils, the ratio 4:1 means for every 4 sharpened pencils, there's 1 blunt one. With 60 pencils total, one part equals 12 pencils $60÷5=12$, so there are 48 sharpened pencils $12×4=48$.

The trick is always the same: find what one part is worth, then multiply to find each person's share.

Quick Check: Your answers should always add up to the original total!

Working Backwards from Ratios

Sometimes you'll need to simplify large number ratios like 1500:9000. Divide both by 1500 to get 1:6 - the Blue Party had 6 times as many votes as the Green Party.

Three-way ratios work identically to two-way ones. For carpet cut in ratio 1:2:5, add the parts $1+2+5=8$, find one part's value $240÷8=30cm$, then find the longest piece $30×5=150cm$.

These problems test whether you can spot the biggest ratio number and calculate accordingly. The longest piece always corresponds to the largest number in the ratio.

Pro Tip: In three-way ratios, identify which part of the ratio you're being asked to find before you start calculating.

Multi-Part Ratio Problems

Three-way chocolate ratios like 24:16:8 simplify by finding the highest common factor (8), giving 3:2:1. Always check your simplification by seeing if the numbers can be reduced further.

When you know one quantity and need to find another, work backwards through the ratio. Rachel has 14 apples in a 2:3 ratio with bananas. Since 14÷2=7, each ratio part represents 7 fruits, so she has 7×3=21 bananas.

This reverse-engineering approach is crucial for many exam questions. You're often given one piece of information and asked to find the rest.

Key Insight: If you know one part of a ratio, you can find the value of each ratio unit and calculate everything else.

Money Problems and Competition Sharing

Ratio money problems often give you one person's share and ask for the other amounts. If Molly gets £240 in a 2:3 ratio, find the value of one part $£240÷3=£80$, then calculate Chris's share $£80×2=£160$ and the total $£240+£160=£400$.

Direct division problems like splitting £945 in ratio 2:5 follow the standard method: total parts = 7, so each part = £135. Chris gets £270 (135×2) and Molly gets £675 (135×5).

These questions test your ability to move between different types of ratio problems smoothly.

Money Saver: Always double-check that individual amounts add up to the original total - it catches most calculation errors!

Advanced Multi-Step Calculations

Complex ratio word problems combine several skills. At the rugby match, 80 children in a 2:3 ratio means 120 adults $80÷2×3=120$. With adult tickets at £8 and child tickets at £2 (quarter of £8), total revenue is £960+£160=£1120.

When someone gets "£25 more" in a ratio problem, find the difference in ratio parts. In ratio 2:3, the difference is 1 part = £25, so Charlene gets £50 (2×£25) and Danielle gets £75 (3×£25).

These multi-step problems reward careful planning and systematic working.

Strategy Tip: Break complex problems into smaller steps and solve each part before moving to the next.

Ratios, Fractions, and Percentages

School population problems test your understanding of ratios as parts of a whole. With 220 boys in a 4:5 ratio, there are 275 girls $220÷4×5=275$ and 495 total students.

Converting ratios to fractions and percentages is straightforward once you grasp the concept. In a 2:3 ratio of girls to boys, total parts = 5, so girls are 2/5 of the class and boys are 3/5 = 60%.

This connection between ratios, fractions, and percentages appears frequently in GCSE questions.

Connection Point: Ratios are just another way of expressing fractions and percentages - they're all showing parts of a whole!

Angles and Geometry Ratios

Triangle angle ratios use the fact that angles sum to 180°. In ratio 1:2:9, total parts = 12, so each part = 15° $180÷12=15$. The largest angle is 135° $15×9=135$.

When given actual angle measurements, work backwards to find the ratio value. If the smallest angle in ratio 2:3:5 is 50°, then each ratio part equals 25° $50÷2=25$, making the other angles 75° and 125°.

Geometry ratio problems combine your ratio skills with basic geometric facts.

Angle Alert: Remember that triangle angles always add to 180° - use this to check your answers make sense!

Complex Multi-School Problems

Advanced ratio problems can involve multiple schools and different constraints. When one school sends French and German students in ratio 1:3, and you know there are 21 German students, you can find there are 7 French students $21÷3×1=7$, totalling 28 students from this school.

If the other three schools send equal numbers, and the total is 112 students, then each of the other schools sent 28 students too (since 28×4=112).

These problems test your ability to handle multiple ratios and constraints simultaneously.

Final Strategy: For complex problems, identify all the given information first, then work through each constraint step by step.