Welcome to Calculations with Ratio!
Ever tried to bake a cake and needed to double the recipe? Or shared a bag of sweets with a friend where one of you got more because you helped out more? That is ratio in action! In this chapter, we are going to learn how to compare quantities, simplify them, and share things out fairly. Don't worry if you find numbers a bit intimidating; we will take this step-by-step.
This chapter is part of the Ratio, Proportion and Rates of Change section of your OCR GCSE (9-1) course.
Did you know? The "Golden Ratio" is a special number found in nature, from the petals of flowers to the shape of galaxies! Maths really is everywhere.
1. Understanding and Simplifying Ratios
What is a Ratio?
A ratio is a way of comparing two or more quantities to show how much of one there is compared to another. We use a colon (:) to separate the numbers. For example, if a recipe uses 2 cups of flour for every 1 cup of sugar, the ratio of flour to sugar is 2 : 1.
Equivalent Ratios
Just like fractions, ratios can be simplified. To simplify a ratio, you divide all the numbers in the ratio by the same Highest Common Factor (HCF).
Example: Simplify the ratio 10 : 15.
1. Find a number that goes into both 10 and 15. That’s 5!
2. \(10 \div 5 = 2\)
3. \(15 \div 5 = 3\)
The simplified ratio is 2 : 3.
The Unit Ratio Form (\(1 : n\))
Sometimes, the exam will ask you to write a ratio in the form 1 : n. This means the first number must be 1, even if the second number becomes a decimal.
Example: Write 5 : 8 in the form 1 : n.
To turn the 5 into a 1, we must divide by 5. We must do the same to the other side!
\(5 \div 5 = 1\)
\(8 \div 5 = 1.6\)
The answer is 1 : 1.6.
Important Rule: Check Your Units!
Before you write a ratio, make sure both quantities are in the same units. If they aren't, convert them first!
Example: Find the ratio of 50 cm to 1.5 m.
1. Convert 1.5 m to cm: \(1.5 \times 100 = 150\) cm.
2. Write as a ratio: 50 : 150.
3. Simplify: Divide both by 50 to get 1 : 3.
Key Takeaway: Always simplify ratios by dividing by a common factor, and always make sure your units match before you start!
2. Sharing a Quantity in a Given Ratio
This is a very common exam question. Imagine you have to share some money between two people, but not equally.
Step-by-Step: The "Add, Divide, Multiply" Method
Let's say we want to share £50 in the ratio 2 : 3.
Step 1: Add the parts.
Find the total number of "shares."
\(2 + 3 = 5\) parts in total.
Step 2: Divide the total amount.
Find out how much one part is worth.
\(£50 \div 5 = £10\) (This is the value of 1 part).
Step 3: Multiply to find the shares.
Person A gets 2 parts: \(2 \times £10 = £20\).
Person B gets 3 parts: \(3 \times £10 = £30\).
Check: \(£20 + £30 = £50\). It adds up!
Sharing into Three or More Parts
The process is exactly the same! If the ratio is 1 : 2 : 3, you just add all three numbers together (\(1+2+3=6\)) and follow the same steps.
Quick Review: To share in a ratio: Add the parts, Divide the total by that sum, then Multiply that answer by the individual ratio numbers.
3. Ratios and Fractions
Ratios and fractions are cousins—they are closely related! You can turn any ratio into a fraction of the whole amount.
If the ratio of Blue pens to Red pens is 3 : 7:
1. The total number of parts is \(3 + 7 = 10\).
2. The fraction of Blue pens is \( \frac{3}{10} \).
3. The fraction of Red pens is \( \frac{7}{10} \).
Common Mistake to Avoid: Many students see the ratio 3 : 7 and think the fraction is \( \frac{3}{7} \). No! The denominator (the bottom number) must always be the total of all the parts added together.
Key Takeaway: Ratio \(a : b\) means the fractions are \( \frac{a}{a+b} \) and \( \frac{b}{a+b} \).
4. Solving Ratio and Proportion Problems
Sometimes you aren't given the total amount; instead, you are told how much one person has and asked to find the rest.
Example: Finding an Unknown Part
The ratio of boys to girls in a club is 4 : 5. If there are 20 boys, how many girls are there?
1. We know 4 parts = 20 boys.
2. Find the value of 1 part: \(20 \div 4 = 5\).
3. The girls have 5 parts, so: \(5 \times 5 = 25\) girls.
Adapting Recipes (Proportion)
This is a classic real-world use of ratio. If a recipe for 4 people uses 200g of flour, how much do you need for 6 people?
1. Find the amount for 1 person: \(200\text{g} \div 4 = 50\text{g}\).
2. Multiply for 6 people: \(50\text{g} \times 6 = 300\text{g}\).
Analogy: Think of the "1 part" value as the "Size of the Slice." Once you know how big one slice is, you can figure out how much any number of slices weighs or costs!
Key Takeaway: Find what "one part" or "one item" is worth first. This is often called the Unitary Method.
Summary Checklist for Your Revision
- Can I simplify a ratio like a fraction?
- Do I remember to check that units (cm, m, kg, g) are the same?
- Can I write a ratio in the form 1 : n?
- Do I know the Add -> Divide -> Multiply steps for sharing?
- Can I turn a ratio into a fraction by adding the parts for the denominator?
Don't worry if this seems tricky at first! Ratio is all about practice. Try starting with simple 1 : 2 ratios and work your way up to the bigger numbers. You've got this!