How to Solve: Ratio Problems

How to Solve: Ratio Problems

(For Students Who Want to Ace Their Exam & Teachers Who Need a Ready-to-Record Script)

Introduction

"Ratios decide how much pizza each friend gets, how much medicine to take, and even how much money you split with your siblings—mess this up, and you’ll lose marks on every math exam from now until graduation!

What You Need To Know First

Before tackling ratios, you must already understand: 1. Division and multiplication – You’ll need to split quantities into parts. 2. Fractions – Ratios can be written as fractions (e.g., 3:4 = 3/4). 3. Basic algebra – Sometimes, you’ll solve for an unknown (e.g., x).

If any of these feel shaky, review them first—ratios build on them!

Key Vocabulary

Formulas To Know

1. Basic Ratio Formula

Formula: Part = (Ratio Part / Total Parts) × Total Quantity

Variables: - Ratio Part = One number in the ratio (e.g., in 3:5, 3 is a ratio part). - Total Parts = Sum of all ratio parts (e.g., 3 + 5 = 8). - Total Quantity = The actual amount being divided.

MEMORISE THIS – You’ll use it in every ratio problem.

2. Scaling Ratios (Equivalent Ratios)

Formula: a : b = (a × k) : (b × k) (where k is a scaling factor)

Example: Scale 2:5 by 3 → (2×3):(5×3) = 6:15.

Given on exam sheet? Sometimes, but memorise it—it’s faster.

3. Combining Ratios

Formula: If A : B = x : y and B : C = y : z, then A : B : C = x : y : z.

Example: If boys:girls = 3:4 and girls:teachers = 4:1, then boys:girls:teachers = 3:4:1.

MEMORISE THIS – Examiners love multi-step ratio problems.

Step-by-Step Method

Follow these steps for every ratio problem. No exceptions.

Step 1: Read the problem. Underline the ratio and the total quantity.

• Example: "A bag has red and blue marbles in the ratio 3:5. There are 40 marbles in total. How many are red?" → Ratio = 3:5, Total = 40.

Step 2: Find the total number of parts.

• Add the ratio numbers: 3 + 5 = 8 parts.

Step 3: Find the value of one part.

• Divide the total quantity by total parts: 40 ÷ 8 = 5. → 1 part = 5 marbles.

Step 4: Multiply the ratio part by the value of one part.

• Red marbles = 3 parts × 5 = 15 marbles.

Step 5: Check your answer.

• Blue marbles = 5 × 5 = 25.
• Total = 15 + 25 = 40 ✔️.

Worked Examples

Example 1 – Basic (Direct Division)

Problem: "A recipe uses flour and sugar in the ratio 2:7. If you use 280g of sugar, how much flour do you need?"

Working: 1. Ratio = 2:7 (flour:sugar). 2. Total parts = 2 + 7 = 9. 3. Sugar = 7 parts = 280g.
→ 1 part = 280 ÷ 7 = 40g. 4. Flour = 2 × 40 = 80g.

Answer: 80g of flour.

What we did and why: - We used the ratio to find the value of one part, then scaled it up for flour. - Always check: 80:280 simplifies to 2:7 ✔️.

Example 2 – Medium (Three-Part Ratio)

Problem: "In a class, the ratio of boys to girls to teachers is 5:3:1. If there are 9 teachers, how many students are there?"

Working: 1. Ratio = 5:3:1 (boys:girls:teachers). 2. Teachers = 1 part = 9.
→ 1 part = 9 ÷ 1 = 9. 3. Boys = 5 × 9 = 45.
Girls = 3 × 9 = 27. 4. Total students = 45 + 27 = 72.

Answer: 72 students.

What we did and why: - We found the value of one part using the known quantity (teachers). - Then, we scaled up for boys and girls. - Trick: Teachers are not students—don’t include them in the final count!

Example 3 – Exam Style (Disguised Ratio)

Problem: "A map scale is 1:50,000. If two towns are 6cm apart on the map, what is the real distance in km?"

Working: 1. Ratio = 1:50,000 (map:real). 2. Map distance = 6cm → Real distance = 6 × 50,000 = 300,000 cm. 3. Convert cm to km:
- 300,000 cm ÷ 100 = 3,000 m.
- 3,000 m ÷ 1,000 = 3 km.

Answer: 3 km.

What we did and why: - The ratio is a scale, not a direct division. - We scaled up the map distance, then converted units. - Exam trap: Always check if the answer needs unit conversion!

Common Mistakes

Exam Traps

1-Minute Recap

"Okay, listen up—this is your 60-second ratio survival guide. First, underline the ratio and the total quantity in the problem. Second, add the ratio parts to get total parts. Third, divide the total quantity by total parts to find one part. Fourth, multiply the ratio part by one part’s value to get your answer. Fifth, check it—does it make sense? If the problem has three parts, combine the ratios first. If it’s a scale, convert units at the end. And whatever you do, don’t mix up the order—boys:girls is not the same as girls:boys. Now go crush that exam!