Introduction to Ratios
Table Of Contents
🎬 Math Angel Video: Introduction to Ratios
What Is a Ratio?​
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🛎️ Definition of A Ratio:
A ratio is a way to compare two or more quantities.
🛎️ Notation of Ratios:
Ratios are written in the form: $$ a : b$$
When writing a ratio, the order of the numbers must match the order of the items.
🛎️ Examples of Ratios:
You have 2 apples and 3 bananas
Ratio of apples to bananas:
$$2 : 3$$
Ratio of bananas to apples:
$$3 : 2$$
Using Ratios to Find an Amount (Example)
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🚨 Question:
If the flour to sugar ratio is 5 : 2, and you have 300 g of flour, how much sugar do you need?
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🛎️ Step 1: Understand what the ratio 5 : 2 means
The ratio 5 : 2 tells us how the two ingredients compare.
- For every 5 parts of flour, there are 2 parts of sugar.
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🛎️ Step 2: Find the value of 1 part of the ratio
- Flour makes up the 5 parts in the ratio, and this equals 300 g.
- So to find the value of 1 part, we divide by 5:
$$\text{1 part} = \frac{300\,\text{g}}{5} = 60\,\text{g}$$
- This tells us that each part in the ratio is 60 g.
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🛎️ Step 3: Find the sugar amount
- Sugar is the 2 parts in the ratio. Each part is 60 g, so:
$$\text{Sugar needed} = 2 \times 60\,\text{g} = 120\,\text{g}$$
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❇️ Final Answer:
You need 120 g of sugar.
How to Share an Amount in a Ratio (Example)
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🚨 Question:
You and a friend want to share ÂŁ200 in the ratio 7 : 3.
How much will each of you get?
🛎️ Step 1: Work out the total number of parts
$$\text{Total parts: } 7 + 3 = 10$$
🛎️ Step 2: Find the value of 1 part
The ÂŁ200 must be shared equally across all 10 parts:
$$\text{1 part} = \frac{\text{ÂŁ}200}{10} = \text{ÂŁ}20$$
🛎️ Step 3: Find each person’s share
- Your share (7 parts): $$7 \times ÂŁ20 = ÂŁ140$$
- Friend’s share (3 parts): $$3 \times £20 = £60$$
How to Simplify and Expand Ratios?
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🛎️ What does “simplifying a ratio” mean?
- Simplifying a ratio means dividing both numbers by the same common factor.
- The numbers become smaller, but the relationship stays the same.
- It is the same idea as simplifying a fraction.
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🛎️ Example: Simplify 24 : 40
- Step 1: Divide both numbers by a common factor (4 is a common factor of both 24 and 40)
$$24 \div 4 = 6,\quad 40 \div 4 = 10$$ - Step 2: Divide again if there is another common factor (2 goes into both 6 and 10)
$$6 \div 2 = 3,\quad 10 \div 2 = 5$$ - Final simplified ratio:
$$24 : 40 = 3 : 5$$
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🛎️ What does “expanding a ratio” mean?
- Expanding a ratio means multiplying both numbers by the same number.
- This makes the numbers larger while keeping the ratio equivalent.
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🛎️ Example: Expand 3 : 7 by 5
Multiply both numbers by 5:
$$(3 \times 5) : (7 \times 5) = 15 : 35$$
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❇️ Exam Tips:
- Whether you simplify (divide) or expand (multiply), you must do the same operation to both numbers.
- This does not change the relationship between the numbers. It only makes the ratio easier to work with for calculations.
🍪 Practice: Calculating, Expanding & Simplifying Ratios
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