A bag holds 5 oranges and 20 apples. The ratio of oranges to apples is 5 : 20, which is the same as 1 : 4. That is the whole idea of a ratio: it compares two quantities.
This article covers ratios, sharing money, proportion, and direct and inverse variation. It ends with five solved examples. Each one says why the method works.
- Ratio: a comparison of two quantities in the same unit.
- Simplest form: divide both terms by their HCF.
- Sharing: add the ratio parts, find one part, then multiply.
- Proportion: a : b = c : d, so a × d = b × c.
- Third proportional: to a and b is b² ÷ a.
- Mean proportional: between a and b is √(a × b).
- Inverse variation: x × y stays constant.
What is a ratio and how do you simplify it?
What is a ratio?
A ratio compares two quantities of the same kind. We write it as a : b.
If a class has 18 girls and 12 boys, the ratio of girls to boys is 18 : 12. In simplest form, it is 3 : 2.
Sources: NCERT Class 8 Maths: Comparing Quantities, recalling ratios.
How do you write a ratio in its simplest form?
Divide both terms by their HCF. The HCF is the highest common factor.
For 102 : 162, the HCF is 6. So the ratio is 17 : 27, and no more cancelling is possible.
Sources: NCERT Class 8 Maths: Comparing Quantities, ratio of girls to boys; Practice question: simplest form of 102:162.
What if the two quantities have different units?
First change both to one unit. Then form the ratio.
Rs 2.5 is 250 paise. So Rs 2.5 : 52 paise = 250 : 52 = 125 : 26.
Sources: NCERT Class 8 Maths: Comparing Quantities, ratios as comparison.
How do you share money in a given ratio?
Add the ratio parts. Divide the total by that sum to get one part. Then multiply by each term.
For Rs 594 in the ratio 2 : 9, there are 11 parts. One part is Rs 54. The shares are Rs 108 and Rs 486.
Does the same method work for three shares?
Yes. Add all three terms. For Rs 396 in the ratio 8 : 7 : 3, the sum is 18.
One part is 396 ÷ 18 = Rs 22. The shares are 176, 154 and 66. The largest is Rs 176.
How do you find the ratio when percentages are equal?
Write both as fractions of one number. If 37% of A equals 20% of B, then 37A = 20B.
So A : B = 20 : 37. The multiplier moves across to the other side.
What are proportion and its special cases?
What is a proportion?
A proportion says two ratios are equal. We write a : b = c : d.
Then the product of the outer terms equals the product of the inner terms. So a × d = b × c.
Sources: NCERT Class 8 Maths: Direct and Inverse Proportions, constant ratio.
How do you find the fourth proportional?
The fourth proportional to a, b, c is d in a : b = c : d. So d = (b × c) ÷ a.
For 6, 13 and 18, d = 13 × 18 ÷ 6 = 39.
Sources: Practice question: fourth proportional to 6, 13, 18.
How do you find the third proportional?
The third proportional to a and b is c in a : b = b : c. So c = b² ÷ a.
For 9 and 12, c = 144 ÷ 9 = 16.
How do you find the mean proportional?
The mean proportional between a and b is m in a : m = m : b. So m² = a × b, and m = √(a × b).
For 2 and 18, m = √36 = 6.
What is a compound ratio?
Multiply the first terms together. Then multiply the second terms together.
The compound ratio of 4 : 3 and 8 : 4 is 32 : 12, which simplifies to 8 : 3.
Check 1
The ratio 360 : 180 simplifies to 2 : 1.
Check 2
The mean proportional between 2 and 18 is 10.
Check 3
The third proportional to 9 and 12 is 16.
Check 4
To share Rs 594 in 2 : 9, divide it by 2.
How do direct and inverse proportion differ?
What is direct proportion?
Two quantities are in direct proportion if both rise or fall together. Their ratio stays constant.
If 5 metres of cloth cost Rs 400, then 8 metres cost Rs 640. The price per metre is always Rs 80.
Sources: NCERT Class 8 Maths: Direct and Inverse Proportions, direct proportion.
What is inverse proportion?
In inverse proportion, one quantity rises while the other falls. Their product stays constant.
So x × y = k. If x1 × y1 = x2 × y2, the two are inversely proportional.
Sources: NCERT Class 8 Maths: Direct and Inverse Proportions, inverse proportion.
How do you solve an inverse proportion problem?
Find the constant product first. Then divide it by the new value.
For 16 workers in 3 days, the product is 48. With 8 workers, days = 48 ÷ 8 = 6.
Sources: NCERT Class 8 Maths: Direct and Inverse Proportions, workers and days; Practice question: 16 workers in 3 days.
How do I tell which kind it is?
Ask one question. If one grows, does the other grow too? If yes, it is direct. If the other shrinks, it is inverse.
More workers means more units made. But more workers means fewer days needed.
Sources: NCERT Class 8 Maths: Direct and Inverse Proportions, comparing the two kinds.
How do you solve five typical problems step by step?
Example 1: Write 84 : 126 in its simplest form.
Step 1. The HCF of 84 and 126 is 42.
Step 2. 84 ÷ 42 = 2, and 126 ÷ 42 = 3.
Step 3. The ratio is 2 : 3.
Check. 2 × 42 = 84 and 3 × 42 = 126.
Why this works. Dividing both terms by one number keeps the comparison the same.
Sources: NCERT Class 8 Maths: Comparing Quantities, equal ratios in simplest form.
Example 2: Share Rs 1,350 in the ratio 4 : 5.
Step 1. Total parts = 4 + 5 = 9.
Step 2. One part = 1350 ÷ 9 = Rs 150.
Step 3. The shares are 4 × 150 = Rs 600 and 5 × 150 = Rs 750.
Check. 600 + 750 = 1350.
Why this works. Each part is equal. So the total is 9 equal parts.
Sources: NCERT Class 8 Maths: Comparing Quantities, ratio of girls to boys.
Example 3: Share Rs 720 in the ratio 1 : 2 : 3.
Step 1. Total parts = 1 + 2 + 3 = 6.
Step 2. One part = 720 ÷ 6 = Rs 120.
Step 3. The shares are Rs 120, Rs 240 and Rs 360.
Check. 120 + 240 + 360 = 720.
Why this works. Each share is a whole number of equal parts.
Sources: NCERT Class 8 Maths: Comparing Quantities, comparing oranges and apples.
Example 4: Find the fourth proportional to 4, 9 and 20.
Step 1. Write 4 : 9 = 20 : d.
Step 2. Cross-multiply: 4 × d = 9 × 20 = 180.
Step 3. So d = 180 ÷ 4 = 45.
Check. 4 × 45 = 180 and 9 × 20 = 180.
Why this works. Equal ratios have equal cross products.
Sources: NCERT Class 8 Maths: Direct and Inverse Proportions, equal ratios across terms.
Example 5: 12 workers finish a job in 10 days. How long will 15 workers take?
Step 1. More workers means fewer days. This is inverse proportion.
Step 2. Workers × days = 12 × 10 = 120.
Step 3. Days = 120 ÷ 15 = 8 days.
Check. 15 × 8 = 120.
Why this works. The total work is fixed. So workers × days is the same each time.
Sources: NCERT Class 8 Maths: Direct and Inverse Proportions, product stays constant.
How do you avoid careless mistakes?
What habits keep answers right?
Put both quantities in one unit before forming a ratio. Write the order of terms and keep it.
After sharing, add the shares. The sum must equal the total.
Check 5
The ratio of Rs 3 to 50 paise is 3 : 50.
Check 6
If 8 workers take 6 days, 16 workers take 3 days.
Exam traps
Unit trap. Rupees and paise must be in one unit before you form a ratio.
Order trap. The ratio of A to B is not the ratio of B to A.
Sharing trap. Divide by the sum of the parts, not by one term.
Direct-or-inverse trap. More workers means fewer days. Check the direction first.
Mean trap. The mean proportional is the square root of the product, not the average.
MCQ on Ratio and Proportion
0 of 1 answered Question 1 of 1 0 of 1 answered Question 1 of 1 0 of 1 answered Question 1 of 1 0 of 1 answered Question 1 of 1 0 of 1 answered Question 1 of 1 0 of 1 answered Question 1 of 1 0 of 1 answered Question 1 of 1
Leave a Reply