SSC GD Ratio and Proportion Notes 2026 – Complete Handwritten Notes, Theory, Formulas & Solved Questions

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SSC GD Ratio & Proportion

Detailed Theory • Easy English • Handwritten Notes • SSC GD 2026

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📌 Clickable Index

1. Ratio Basics2. Equivalent Ratios3. Simplifying Ratios4. Comparing Ratios5. Division in a Ratio6. Missing Term7. Compound Ratio8. Proportion9. Fourth Proportional10. Third Proportional11. Mean Proportional12. Direct Proportion13. Inverse Proportion14. Ratio as Percentage15. Mixture Ratio16. 10 Fully Solved Questions17. Final Revision

Chapter idea: Ratio compares quantities. Proportion tells us that two ratios are equal. The same methods are used in sharing, ages, prices, work, speed and mixtures.

1. Ratio – Meaning & Basics

Theory

A ratio compares two quantities of the same kind by division. It is written using a colon. In a ratio a:b, a is the first term and b is the second term.

\[a:b=\frac ab,\qquad b\ne0\]

Order matters: boys:girls is different from girls:boys. Also, units should be the same before forming a ratio.

Example: 2 m and 50 cm. Convert 2 m = 200 cm. Ratio = 200:50 = 4:1.

3 Step-by-Step Questions

Q1. Find the ratio of 18 to 24.

1. Write 18:24.
2. HCF of 18 and 24 = 6.
3. Divide both terms by 6: 18÷6 : 24÷6 = 3:4.
4. Answer: 3:4.

Q2. Find the ratio of 3 kg to 750 g.

1. Convert 3 kg = 3000 g.
2. Ratio = 3000:750.
3. Divide by 750.
4. Answer: 4:1.

Q3. Boys:girls = 5:7. If boys are 25, find girls.

1. 5 parts = 25.
2. 1 part = 25÷5 = 5.
3. Girls = 7×5.
4. Answer: 35 girls.

2. Equivalent Ratios

Theory

Equivalent ratios have the same value. We get them by multiplying or dividing both terms by the same non-zero number.

\[a:b=ka:kb\]

Example: 4:9 = 8:18 = 12:27.

3 Step-by-Step Questions

Q1. Write an equivalent ratio of 6:11 by ×4.

1. Multiply first term: 6×4=24.
2. Multiply second term: 11×4=44.
3. Answer: 24:44.

Q2. Check 15:25 and 3:5.

1. Simplify 15:25.
2. HCF = 5.
3. 15÷5:25÷5 = 3:5.
4. Answer: Yes, they are equivalent.

Q3. 7:9 = x:27.

1. Write 7/9 = x/27.
2. Cross multiply: 9x = 7×27.
3. x = 189÷9 = 21.
4. Answer: x=21.

3. Simplifying a Ratio

Theory

To reduce a ratio to lowest form, divide both terms by their HCF. For three terms, divide all terms by their common HCF.

3 Step-by-Step Questions

Q1. Simplify 48:72.

1. HCF of 48 and 72 = 24.
2. 48÷24=2.
3. 72÷24=3.
4. Answer: 2:3.

Q2. Simplify 84:126.

1. HCF = 42.
2. 84÷42=2.
3. 126÷42=3.
4. Answer: 2:3.

Q3. Simplify 24:36:60.

1. HCF of 24,36,60 = 12.
2. 24÷12=2, 36÷12=3, 60÷12=5.
3. Answer: 2:3:5.

4. Comparing Ratios

Theory

To compare a:b and c:d, compare a/b and c/d. Cross multiplication is usually fastest.

\[a:b>c:d\iff ad>bc\]

Q1. Compare 2:3 and 3:5.

1. Cross products: 2×5=10.
2. 3×3=9.
3. 10>9.
4. Answer: 2:3 is greater.

Q2. Compare 5:8 and 7:11.

1. 5×11=55.
2. 7×8=56.
3. 56>55.
4. Answer: 7:11 is greater.

Q3. Compare 4:9 and 6:13.

1. 4×13=52.
2. 6×9=54.
3. 54>52.
4. Answer: 6:13 is greater.

5. Division in a Given Ratio

Theory

If total T is divided in ratio a:b, then total parts = a+b. One part = T/(a+b).

\[\text{First share}=\frac{a}{a+b}T,\quad \text{Second share}=\frac{b}{a+b}T\]

Q1. Divide 540 in 4:5.

1. Total parts = 4+5=9.
2. One part = 540÷9=60.
3. Shares = 4×60 and 5×60.
4. Answer: 240 and 300.

Q2. Divide ₹900 in 2:3.

1. Total parts = 5.
2. One part = ₹900÷5=₹180.
3. Shares = 2×180 and 3×180.
4. Answer: ₹360 and ₹540.

Q3. ₹1200 is shared among A and B in 5:7.

1. Total parts = 12.
2. One part = 1200÷12=100.
3. A=5×100=500; B=7×100=700.
4. Answer: ₹500 and ₹700.

6. Missing Term in a Ratio

Theory

\[\frac ab=\frac cd\Rightarrow ad=bc\]

Cross multiplication removes the fraction and gives a simple equation.

Q1. 5:8 = x:40.

1. 5/8=x/40.
2. 8x=5×40=200.
3. x=25.
4. Answer: 25.

Q2. 9:4 = 45:x.

1. 9/4=45/x.
2. 9x=180.
3. x=20.
4. Answer: 20.

Q3. x:12 = 5:6.

1. x/12=5/6.
2. 6x=60.
3. x=10.
4. Answer: 10.

7. Compound Ratio

Theory

To find the compound ratio of a:b and c:d, multiply corresponding terms.

\[(a:b)\text{ and }(c:d)\Rightarrow ac:bd\]

Q1. Compound ratio of 2:3 and 5:7.

1. Multiply first terms: 2×5=10.
2. Multiply second terms: 3×7=21.
3. Answer: 10:21.

Q2. Compound ratio of 3:4 and 8:9.

1. 3×8=24.
2. 4×9=36.
3. Simplify 24:36 = 2:3.
4. Answer: 2:3.

Q3. Compound ratio of 5:6 and 3:10.

1. 5×3=15.
2. 6×10=60.
3. 15:60 = 1:4.
4. Answer: 1:4.

8. Proportion – Meaning

Theory

Four numbers are in proportion when the first ratio equals the second ratio.

\[a:b=c:d\iff ad=bc\]

Q1. Check 3:5::12:20.

1. 3×20=60.
2. 5×12=60.
3. Both products are equal.
4. Answer: Yes.

Q2. Check 4:7::16:28.

1. 4×28=112.
2. 7×16=112.
3. Answer: Yes.

Q3. Check 5:8::20:30.

1. 5×30=150.
2. 8×20=160.
3. Products differ.
4. Answer: No.

9. Fourth Proportional

Theory

If a:b = c:x, then x is the fourth proportional.

\[x=\frac{bc}{a}\]

Q1. Fourth proportional to 2,5,8.

1. x=5×8÷2.
2. x=40÷2.
3. Answer: 20.

Q2. Fourth proportional to 3,4,9.

1. x=4×9÷3.
2. x=36÷3=12.
3. Answer: 12.

Q3. Fourth proportional to 5,7,15.

1. x=7×15÷5.
2. x=105÷5=21.
3. Answer: 21.

10. Third Proportional

Theory

If a:b = b:x, then x is the third proportional to a and b.

\[x=\frac{b^2}{a}\]

Q1. Third proportional to 2 and 6.

1. x=6²÷2.
2. x=36÷2=18.
3. Answer: 18.

Q2. Third proportional to 5 and 10.

1. x=10²÷5.
2. x=100÷5=20.
3. Answer: 20.

Q3. Third proportional to 4 and 8.

1. x=8²÷4.
2. x=64÷4=16.
3. Answer: 16.

11. Mean Proportional

Theory

The mean proportional x between a and b satisfies a:x = x:b. Therefore x²=ab.

\[x=\sqrt{ab}\]

Q1. Mean proportional between 4 and 9.

1. x=√(4×9).
2. x=√36.
3. Answer: 6.

Q2. Mean proportional between 9 and 16.

1. x=√144.
2. x=12.
3. Answer: 12.

Q3. Mean proportional between 6 and 24.

1. x=√(144).
2. x=12.
3. Answer: 12.

12. Direct Proportion

Theory

When two quantities increase or decrease together in the same ratio, they are directly proportional.

\[y\propto x\Rightarrow\frac yx=k\]

Q1. 5 books cost ₹150. Find 8 books.

1. Cost per book = 150÷5=₹30.
2. Cost of 8 = 8×30.
3. Answer: ₹240.

Q2. 6 kg rice costs ₹300. Find 10 kg.

1. 1 kg = ₹50.
2. 10 kg = 10×50.
3. Answer: ₹500.

Q3. 4 pens cost ₹60. Find 15 pens.

1. 1 pen = ₹15.
2. 15 pens = 15×15.
3. Answer: ₹225.

13. Inverse Proportion

Theory

When one quantity increases while the other decreases and their product remains constant, they are inversely proportional.

\[y\propto\frac1x\Rightarrow xy=k\]

Q1. 8 workers take 15 days. How many days for 12 workers?

1. Workers×days is constant.
2. 8×15=12×x.
3. x=120÷12=10.
4. Answer: 10 days.

Q2. A car takes 6 h at 50 km/h. Time at 75 km/h?

1. Speed×time is constant for the same distance.
2. 50×6=75×x.
3. x=300÷75=4.
4. Answer: 4 hours.

Q3. 10 taps fill a tank in 4 h. 8 taps take?

1. 10×4=8×x.
2. 40=8x.
3. x=5.
4. Answer: 5 hours.

14. Ratio as Percentage

Theory

To find the percentage represented by one part of a ratio, divide that part by the sum of all ratio parts and multiply by 100.

\[\text{Share percent}=\frac{\text{part}}{\text{total parts}}\times100\%\]

Q1. Ratio 2:3. Find first percentage.

1. Total parts = 5.
2. First share = 2/5×100.
3. Answer: 40%.

Q2. Ratio 4:1. Find second percentage.

1. Total parts = 5.
2. Second share = 1/5×100.
3. Answer: 20%.

Q3. Ratio 5:3. Find first percentage.

1. Total parts = 8.
2. First share = 5/8×100.
3. Answer: 62.5%.

15. Mixture & Ratio

Theory

In simple mixture questions, first add the ratio parts and then find each ingredient from the total quantity.

Q1. Milk:water = 5:2 in 28 L.

1. Total parts = 7.
2. Milk = 5/7×28=20 L.
3. Water = 2/7×28=8 L.
4. Answer: 20 L milk, 8 L water.

Q2. Tea:milk = 3:2 in 25 L.

1. Total parts = 5.
2. Tea = 3/5×25=15 L.
3. Milk = 2/5×25=10 L.
4. Answer: 15 L and 10 L.

Q3. Alloy A:B = 2:3 in 40 kg.

1. Total parts = 5.
2. A = 2/5×40=16 kg.
3. B = 3/5×40=24 kg.
4. Answer: 16 kg and 24 kg.

16. 10 Fully Solved SSC GD Questions

Question 1. Divide ₹1800 in the ratio 5:7.

1. Total parts = 5+7=12.
2. One part = 1800÷12=150.
3. First share = 5×150=750.
4. Second share = 7×150=1050.
5. Answer: ₹750 and ₹1050.

Question 2. Find x if 7:9=x:45.

1. 7/9=x/45.
2. 9x=7×45=315.
3. x=315÷9=35.
4. Answer: x=35.

Question 3. If A:B=3:5 and A=24, find B.

1. 3 parts = 24.
2. 1 part = 24÷3=8.
3. B = 5×8=40.
4. Answer: B=40.

Question 4. Check whether 8:12 and 10:15 are proportional.

1. 8/12=2/3.
2. 10/15=2/3.
3. Both ratios are equal.
4. Answer: Yes.

Question 5. Find the fourth proportional to 4, 6, 14.

1. x=6×14÷4.
2. x=84÷4.
3. Answer: 21.

Question 6. Find the third proportional to 3 and 9.

1. x=9²÷3.
2. x=81÷3.
3. Answer: 27.

Question 7. Find the mean proportional between 16 and 25.

1. x=√(16×25).
2. x=√400.
3. Answer: 20.

Question 8. 12 workers finish a job in 10 days. How many days will 15 workers take?

1. Workers×days = constant.
2. 12×10=15×x.
3. x=120÷15=8.
4. Answer: 8 days.

Question 9. 7 books cost ₹420. Find the cost of 12 books.

1. Cost per book = 420÷7=60.
2. Cost of 12 books = 12×60.
3. Answer: ₹720.

Question 10. Milk:water = 4:1 in 45 L. Find milk and water.

1. Total parts = 5.
2. One part = 45÷5=9 L.
3. Milk = 4×9=36 L.
4. Water = 1×9=9 L.
5. Answer: 36 L milk and 9 L water.

17. Final Revision

\[a:b=\frac ab\]
\[a:b=c:d\Rightarrow ad=bc\]
\[\text{Share}=\frac{\text{part}}{\text{total parts}}\times\text{total}\]
\[\text{Fourth proportional}=\frac{bc}{a}\]
\[\text{Third proportional}=\frac{b^2}{a}\]
\[\text{Mean proportional}=\sqrt{ab}\]
\[\text{Direct: }y/x=k\qquad\text{Inverse: }xy=k\]

Exam checklist: Keep units the same, preserve ratio order, use HCF for simplification, add parts before sharing, and identify direct versus inverse proportion before calculating.

Common mistakes: changing the order of the ratio, dividing only one term, forgetting total parts, and using direct proportion in a worker-days or speed-time question where inverse proportion applies.

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SSC GD Maths • Ratio & Proportion Notes