🗺️ Chapter Roadmap
1 What is a Ratio?
A ratio is simply a way to compare two or more quantities. It tells us how much of one thing there is compared to another. It is written using two dots (:) and is actually just another way to write a fraction (division).
Cake Recipe Example:
To bake a perfect cake, a chef needs exactly 2 cups of Flour and 1 cup of Sugar.
2 Dividing Amounts in a Ratio
This is a highly repeated topic in exams. If you have a total amount of money and need to share it between two people in a specific ratio, follow these steps:
📝 Exam Example:
Divide $1000 between Ali and Asad in the ratio of 2 : 3. How much will Asad get?
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Step 1: Find Total Parts
Add the ratio numbers together: 2 + 3 = 5 Total Parts -
Step 2: Find the Value of ONE Part
Divide the total money by the total parts: 1000 ÷ 5 = $200 (Value of 1 part) -
Step 3: Multiply for the Answer
Asad's ratio is 3. So, multiply 3 by the value of one part:
3 × 200 = $600
3 Proportion (The Master Formula)
When two ratios are perfectly equal to each other, they form a Proportion. Its symbol is four dots (::) or simply an equal sign (=).
✨ The Law of Proportion
Called Extremes
Called Means
B × C = A × D
Means (Middle numbers): 3 and 4
Extremes (Outer numbers): 2 and x
Product of Means = Product of Extremes
3 × 4 = 2 × x
12 = 2x
x = 6
4 & 5 Direct vs. Inverse Proportion
This is the most critical part of word problems. You must read the question and decide if it is a Direct or Inverse relationship before solving it.
📈 Direct Proportion
The "Up-Up" Rule: If one quantity increases, the other also increases. If one decreases, the other decreases.
↑ More Money = ↑ More Items bought
↓ Less Petrol = ↓ Less Distance traveled
If 5 pens cost $50, how much do 8 pens cost?
5 pens → $50
8 pens → $x
(Cross multiply: 5x = 400 → x = $80)
⚖️ Inverse Proportion
The "Up-Down" Rule: If one quantity increases, the other decreases (Reverse effect).
↑ More Workers = ↓ Less Time to finish job
↑ Faster Speed = ↓ Less Time to reach destination
If 10 men build a wall in 6 days, how many days will 5 men take?
10 men × 6 days = 60 (Total Work)
5 men × x days = 60
(Straight multiply: 5x = 60 → x = 12 days)