Back to Courses

Quantitative Aptitude · Ratio and Proportion

Ratio- Formulas , Key Points and Examples

Explanation

🔵 RATIO 🟢 1. BASIC CONCEPT OF RATIO A ratio is a comparison of two quantities of the same kind. The ratio of a to b is written as:
a:b a:b
It can also be written as:
ab \frac{a}{b}
Here, a is called the first term or antecedent, and b is called the second term or consequent. 🔴 IMPORTANT The quantities being compared must have the same units before forming a ratio. 🟢 2. SIMPLIFYING A RATIO To simplify a ratio, divide both terms by their greatest common factor.
a:b=ak:bk a:b = \frac{a}{k}:\frac{b}{k}
where k is a common factor of a and b. 🟢 3. EQUIVALENT RATIOS Multiplying or dividing both terms of a ratio by the same non-zero number does not change the ratio.
a:b=ka:kb a:b = ka:kb
where:
k≠0 k\neq0
🟢 4. RATIO IN SAME UNITS Before comparing quantities, convert them into the same unit. For example:
2 m=200 cm 2\text{ m}=200\text{ cm}
Therefore:
2 m:50 cm=200:50=4:1 2\text{ m}:50\text{ cm} = 200:50 = 4:1
🟢 5. RATIO OF THREE OR MORE QUANTITIES Ratios can compare three or more quantities. For example:
a:b:c a:b:c
All quantities must be expressed in the same units. 🟢 6. DIVIDING A QUANTITY IN A GIVEN RATIO If a quantity X is divided in the ratio:
a:b a:b
then the total number of parts is:
a+b a+b
First part:
aa+b×X \frac{a}{a+b}\times X
Second part:
ba+b×X \frac{b}{a+b}\times X
🟢 7. DIVIDING A QUANTITY IN THE RATIO a:b:c If X is divided in the ratio:
a:b:c a:b:c
Total parts:
a+b+c a+b+c
The three parts are:
aXa+b+c \frac{aX}{a+b+c}
bXa+b+c \frac{bX}{a+b+c}
cXa+b+c \frac{cX}{a+b+c}
🟢 8. FINDING THE TOTAL FROM A RATIO If two quantities are in the ratio:
a:b a:b
and their total is X, then:
First Quantity=aa+b×X \text{First Quantity} = \frac{a}{a+b}\times X
Second Quantity=ba+b×X \text{Second Quantity} = \frac{b}{a+b}\times X
🟢 9. FINDING THE RATIO FROM TWO QUANTITIES If two quantities are x and y:
Ratio=x:y \text{Ratio} = x:y
To simplify, divide both terms by their greatest common factor. 🟢 10. RATIO WHEN DIFFERENCE IS GIVEN If two quantities are in the ratio:
a:b a:b
and their difference is D, then:
Difference in parts=b−a \text{Difference in parts} = b-a
Therefore, one part is:
Db−a \frac{D}{b-a}
The quantities are:
aDb−a \frac{aD}{b-a}
and
bDb−a \frac{bD}{b-a}
🟢 11. FINDING RATIO FROM SUM AND DIFFERENCE If two quantities have sum S and difference D:
Larger Quantity=S+D2 \text{Larger Quantity} = \frac{S+D}{2}
Smaller Quantity=S−D2 \text{Smaller Quantity} = \frac{S-D}{2}
Therefore:
Ratio=(S+D):(S−D) \text{Ratio} = (S+D):(S-D)
🟢 12. COMPARING TWO RATIOS To compare:
a:b a:b
and
c:d c:d
compare:
ad ad
and
bc bc
If:
ad>bc ad>bc
then:
a:b>c:d a:b>c:d
🟢 13. COMPOUND RATIO The compound ratio of:
a:b a:b
and:
c:d c:d
is obtained by multiplying corresponding terms.
Compound Ratio=ac:bd \text{Compound Ratio} = ac:bd
🟢 14. DUPLICATE RATIO The duplicate ratio of:
a:b a:b
is:
a2:b2 a^2:b^2
🟢 15. TRIPLICATE RATIO The triplicate ratio of:
a:b a:b
is:
a3:b3 a^3:b^3
🟢 16. SUB-DUPLICATE RATIO The sub-duplicate ratio of:
a:b a:b
is:
a:b \sqrt{a}:\sqrt{b}
🟢 17. SUB-TRIPLICATE RATIO The sub-triplicate ratio of:
a:b a:b
is:
a3:b3 \sqrt[3]{a}:\sqrt[3]{b}
🟢 18. DIRECT PROPORTION Two quantities are in direct proportion when an increase in one quantity causes a proportional increase in the other.
x∝y x\propto y
Therefore:
xy=k \frac{x}{y}=k
or:
x1y1=x2y2 \frac{x_1}{y_1} = \frac{x_2}{y_2}
🟢 19. INVERSE PROPORTION Two quantities are in inverse proportion when an increase in one quantity causes a proportional decrease in the other.
x∝1y x\propto\frac{1}{y}
Therefore:
xy=k xy=k
or:
x1y1=x2y2 x_1y_1=x_2y_2
🟢 20. RATIO AND PROPORTION A proportion states that two ratios are equal.
a:b=c:d a:b=c:d
This can be written as:
ab=cd \frac{a}{b}=\frac{c}{d}
Using cross multiplication:
ad=bc ad=bc
🟢 21. FOURTH PROPORTIONAL If:
a:b=c:x a:b=c:x
then x is the fourth proportional. Using cross multiplication:
ax=bc ax=bc
Therefore:
x=bca x=\frac{bc}{a}
🟢 22. THIRD PROPORTIONAL If a, b and c are in continued proportion:
a:b=b:c a:b=b:c
then:
b2=ac b^2=ac
Therefore:
c=b2a c=\frac{b^2}{a}
🟢 23. MEAN PROPORTIONAL If x is the mean proportional between a and b:
a:x=x:b a:x=x:b
Therefore:
x2=ab x^2=ab
Hence:
x=ab x=\sqrt{ab}
🟢 24. CONTINUED PROPORTION If:
a:b=b:c a:b=b:c
then a, b and c are said to be in continued proportion. The important relation is:
b2=ac b^2=ac
🟢 25. COMPARING RATIOS AFTER CHANGES If two quantities are increased or decreased by different percentages, first calculate their new values and then form the new ratio. For an increase of R%:
New Value=Original Value(1+R100) \text{New Value} = \text{Original Value} \left(1+\frac{R}{100}\right)
For a decrease of R%:
New Value=Original Value(1−R100) \text{New Value} = \text{Original Value} \left(1-\frac{R}{100}\right)
🟢 26. RATIO AND PERCENTAGE If two quantities are in the ratio:
a:b a:b
then the first quantity as a percentage of the total is:
aa+b×100 \frac{a}{a+b}\times100
The second quantity as a percentage of the total is:
ba+b×100 \frac{b}{a+b}\times100
🟢 27. RATIO OF INCOME AND EXPENDITURE If income and expenditure are in the ratio:
a:b a:b
then savings are:
a−b a-b
Therefore:
Savings Ratio=(a−b):b \text{Savings Ratio} = (a-b):b
when expenditure is represented by b. 🟢 28. RATIO IN MIXTURES If two quantities are mixed in the ratio:
a:b a:b
then the total mixture consists of:
a+b a+b
parts. First component:
aa+b×Total Quantity \frac{a}{a+b}\times\text{Total Quantity}
Second component:
ba+b×Total Quantity \frac{b}{a+b}\times\text{Total Quantity}
🟡 KEY POINTS 1. Ratio compares quantities of the same kind. 2. Always convert quantities into the same units before forming a ratio. 3. A ratio can be multiplied or divided by the same non-zero number without changing its value. 4. In a:b, a is the antecedent and b is the consequent. 5. When dividing a quantity in a:b, total parts are a+b. 6. For three quantities in a:b:c, total parts are a+b+c. 7. Direct proportion means both quantities change in the same direction. 8. Inverse proportion means one quantity increases while the other decreases. 9. In a proportion, cross multiplication can be used. 10. Mean proportional between a and b is √ab. 11. In continued proportion:
b2=ac b^2=ac
12. Duplicate ratio means squaring both terms. 13. Triplicate ratio means cubing both terms. 14. Compound ratio is obtained by multiplying corresponding terms. 🔴 EXAM TIP Whenever a ratio problem gives a total, think in terms of "total parts". Whenever a ratio problem gives a difference, think in terms of "difference in parts". For proportion problems, cross multiplication is usually the quickest method.

Example

🟠 EXAMPLE 1: SIMPLIFYING A RATIO QUESTION: Simplify the ratio 24:36. 🟢 SOLUTION: Find the HCF of 24 and 36:
HCF⁡(24,36)=12 \operatorname{HCF}(24,36)=12
Divide both terms by 12:
2412:3612 \frac{24}{12}:\frac{36}{12}
=2:3 =2:3
✅ ANSWER:
2:3 2:3
🟠 EXAMPLE 2: RATIO OF TWO NUMBERS QUESTION: Find the ratio of 45 to 60. 🟢 SOLUTION:
45:60 45:60
Divide both terms by their HCF:
HCF⁡(45,60)=15 \operatorname{HCF}(45,60)=15
4515:6015 \frac{45}{15}:\frac{60}{15}
=3:4 =3:4
✅ ANSWER:
3:4 3:4
🟠 EXAMPLE 3: FINDING AN UNKNOWN TERM QUESTION: If 5:8 = x:40, find x. 🟢 SOLUTION:
58=x40 \frac{5}{8}=\frac{x}{40}
Cross multiply:
8x=5×40 8x=5\times40
8x=200 8x=200
x=25 x=25
✅ ANSWER:
25 25
🟠 EXAMPLE 4: EQUIVALENT RATIOS QUESTION: Find the number that should replace x:
7:9=x:27 7:9=x:27
🟢 SOLUTION:
79=x27 \frac{7}{9}=\frac{x}{27}
Cross multiply:
9x=7×27 9x=7\times27
9x=189 9x=189
x=21 x=21
✅ ANSWER:
21 21
🟠 EXAMPLE 5: RATIO OF BOYS AND GIRLS QUESTION: In a class, there are 18 boys and 12 girls. Find the ratio of boys to girls. 🟢 SOLUTION:
18:12 18:12
The HCF of 18 and 12 is 6.
186:126 \frac{18}{6}:\frac{12}{6}
=3:2 =3:2
✅ ANSWER:
3:2 3:2
🟠 EXAMPLE 6: RATIO OF THREE NUMBERS QUESTION: Find the simplest ratio of 24, 36 and 60. 🟢 SOLUTION: The HCF of 24, 36 and 60 is:
12 12
Divide each number by 12:
2412:3612:6012 \frac{24}{12}:\frac{36}{12}:\frac{60}{12}
=2:3:5 =2:3:5
✅ ANSWER:
2:3:5 2:3:5
🟠 EXAMPLE 7: DIVIDING AN AMOUNT IN A RATIO QUESTION: Divide ₹800 in the ratio 3:5. 🟢 SOLUTION: Total parts:
3+5=8 3+5=8
Value of one part:
8008=100 \frac{800}{8}=100
First share:
3×100=300 3\times100=300
Second share:
5×100=500 5\times100=500
✅ ANSWER:
₹300 and ₹500 ₹300\text{ and }₹500
🟠 EXAMPLE 8: DIVIDING AN AMOUNT IN A RATIO QUESTION: Divide ₹1,200 among A and B in the ratio 2:3. 🟢 SOLUTION: Total parts:
2+3=5 2+3=5
Value of one part:
12005=240 \frac{1200}{5}=240
A's share:
2×240=480 2\times240=480
B's share:
3×240=720 3\times240=720
✅ ANSWER:
₹480 and ₹720 ₹480\text{ and }₹720
🟠 EXAMPLE 9: FINDING TWO NUMBERS FROM THEIR RATIO QUESTION: Two numbers are in the ratio 3:5 and their sum is 64. Find the numbers. 🟢 SOLUTION: Total parts:
3+5=8 3+5=8
Value of one part:
648=8 \frac{64}{8}=8
First number:
3×8=24 3\times8=24
Second number:
5×8=40 5\times8=40
✅ ANSWER:
24 and 40 24\text{ and }40
🟠 EXAMPLE 10: DIFFERENCE GIVEN QUESTION: Two numbers are in the ratio 4:7. Their difference is 24. Find the numbers. 🟢 SOLUTION: Difference in ratio parts:
7−4=3 7-4=3
Three parts represent 24. Therefore, one part is:
243=8 \frac{24}{3}=8
First number:
4×8=32 4\times8=32
Second number:
7×8=56 7\times8=56
✅ ANSWER:
32 and 56 32\text{ and }56
🟠 EXAMPLE 11: RATIO AFTER ADDING A NUMBER QUESTION: Two numbers are in the ratio 3:5. If 8 is added to both numbers, the ratio becomes 5:7. Find the numbers. 🟢 SOLUTION: Let the numbers be:
3x and 5x 3x\text{ and }5x
According to the question:
3x+85x+8=57 \frac{3x+8}{5x+8}=\frac{5}{7}
Cross multiply:
7(3x+8)=5(5x+8) 7(3x+8)=5(5x+8)
21x+56=25x+40 21x+56=25x+40
16=4x 16=4x
x=4 x=4
Therefore, the numbers are:
3(4)=12 3(4)=12
and:
5(4)=20 5(4)=20
✅ ANSWER:
12 and 20 12\text{ and }20
🟠 EXAMPLE 12: RATIO AFTER SUBTRACTING A NUMBER QUESTION: Two numbers are in the ratio 5:7. If 6 is subtracted from each number, the ratio becomes 2:3. Find the numbers. 🟢 SOLUTION: Let the numbers be:
5x and 7x 5x\text{ and }7x
Then:
5x−67x−6=23 \frac{5x-6}{7x-6}=\frac{2}{3}
Cross multiply:
3(5x−6)=2(7x−6) 3(5x-6)=2(7x-6)
15x−18=14x−12 15x-18=14x-12
x=6 x=6
Therefore:
5x=30 5x=30
and:
7x=42 7x=42
✅ ANSWER:
30 and 42 30\text{ and }42
🟠 EXAMPLE 13: RATIO OF AGES QUESTION: The ages of A and B are in the ratio 3:4. If their total age is 42 years, find their ages. 🟢 SOLUTION: Total parts:
3+4=7 3+4=7
One part:
427=6 \frac{42}{7}=6
A's age:
3×6=18 3\times6=18
B's age:
4×6=24 4\times6=24
✅ ANSWER:
18 years and 24 years 18\text{ years and }24\text{ years}
🟠 EXAMPLE 14: AGE RATIO AFTER SOME YEARS QUESTION: The present ages of A and B are in the ratio 2:3. After 5 years, their ages will be in the ratio 3:4. Find their present ages. 🟢 SOLUTION: Let their present ages be:
2x and 3x 2x\text{ and }3x
After 5 years:
2x+5 2x+5
and:
3x+5 3x+5
According to the question:
2x+53x+5=34 \frac{2x+5}{3x+5}=\frac{3}{4}
Cross multiply:
4(2x+5)=3(3x+5) 4(2x+5)=3(3x+5)
8x+20=9x+15 8x+20=9x+15
x=5 x=5
Therefore:
2x=10 2x=10
and:
3x=15 3x=15
✅ ANSWER:
10 years and 15 years 10\text{ years and }15\text{ years}
🟠 EXAMPLE 15: RATIO OF MONEY QUESTION: A and B have money in the ratio 5:7. If B has ₹840, how much money does A have? 🟢 SOLUTION:
AB=57 \frac{A}{B}=\frac{5}{7}
Therefore:
A840=57 \frac{A}{840}=\frac{5}{7}
7A=5×840 7A=5\times840
7A=4200 7A=4200
A=600 A=600
✅ ANSWER:
₹600 ₹600
🟠 EXAMPLE 16: RATIO OF LENGTHS QUESTION: The lengths of two ropes are in the ratio 7:9. If the shorter rope is 35 m long, find the length of the longer rope. 🟢 SOLUTION:
79=35x \frac{7}{9}=\frac{35}{x}
Cross multiply:
7x=35×9 7x=35\times9
7x=315 7x=315
x=45 x=45
✅ ANSWER:
45 m 45\text{ m}
🟠 EXAMPLE 17: COMPOUND RATIO QUESTION: Find the compound ratio of 2:3 and 4:5. 🟢 SOLUTION: Multiply the corresponding terms:
2×4:3×5 2\times4:3\times5
=8:15 =8:15
✅ ANSWER:
8:15 8:15
🟠 EXAMPLE 18: DUPLICATE RATIO QUESTION: Find the duplicate ratio of 3:5. 🟢 SOLUTION: Square both terms:
32:52 3^2:5^2
=9:25 =9:25
✅ ANSWER:
9:25 9:25
🟠 EXAMPLE 19: TRIPLICATE RATIO QUESTION: Find the triplicate ratio of 2:3. 🟢 SOLUTION: Cube both terms:
23:33 2^3:3^3
=8:27 =8:27
✅ ANSWER:
8:27 8:27
🟠 EXAMPLE 20: SUB-DUPLICATE RATIO QUESTION: Find the sub-duplicate ratio of 25:49. 🟢 SOLUTION: Take the square root of both terms:
25:49 \sqrt{25}:\sqrt{49}
=5:7 =5:7
✅ ANSWER:
5:7 5:7
🟠 EXAMPLE 21: RATIO OF NUMBERS WITH HCF QUESTION: The ratio of two numbers is 7:9 and their HCF is 6. Find the numbers. 🟢 SOLUTION: Let the numbers be:
7x and 9x 7x\text{ and }9x
Since 7 and 9 are relatively prime:
x=6 x=6
Therefore:
7x=42 7x=42
and:
9x=54 9x=54
✅ ANSWER:
42 and 54 42\text{ and }54
🟠 EXAMPLE 22: RATIO AND TOTAL QUESTION: The ratio of boys to girls in a class is 5:3. If there are 40 boys, find the number of girls. 🟢 SOLUTION:
53=40x \frac{5}{3}=\frac{40}{x}
Cross multiply:
5x=40×3 5x=40\times3
5x=120 5x=120
x=24 x=24
✅ ANSWER:
24 girls 24\text{ girls}
🟠 EXAMPLE 23: RATIO OF INCOMES QUESTION: The incomes of A and B are in the ratio 4:5. If B earns ₹30,000, find A's income. 🟢 SOLUTION:
45=x30000 \frac{4}{5}=\frac{x}{30000}
5x=4×30000 5x=4\times30000
5x=120000 5x=120000
x=24000 x=24000
✅ ANSWER:
₹24,000 ₹24,000
🟠 EXAMPLE 24: RATIO OF EXPENDITURE QUESTION: The incomes of A and B are in the ratio 5:6 and their expenditures are in the ratio 3:4. If A saves ₹2,000 and B saves ₹1,000, find their incomes. 🟢 SOLUTION: Let their incomes be:
5x and 6x 5x\text{ and }6x
Let their expenditures be:
3y and 4y 3y\text{ and }4y
For A:
5x−3y=2000 5x-3y=2000
For B:
6x−4y=1000 6x-4y=1000
Multiply the first equation by 4:
20x−12y=8000 20x-12y=8000
Multiply the second equation by 3:
18x−12y=3000 18x-12y=3000
Subtract:
2x=5000 2x=5000
x=2500 x=2500
Therefore:
5x=12500 5x=12500
and:
6x=15000 6x=15000
✅ ANSWER:
₹12,500 and ₹15,000 ₹12,500\text{ and }₹15,000
🟠 EXAMPLE 25: RATIO OF MIXTURE QUESTION: Milk and water are mixed in the ratio 5:2. If the total mixture is 28 litres, find the quantity of milk and water. 🟢 SOLUTION: Total parts:
5+2=7 5+2=7
One part:
287=4 \frac{28}{7}=4
Milk:
5×4=20 5\times4=20
Water:
2×4=8 2\times4=8
✅ ANSWER:
20 L milk and 8 L water 20\text{ L milk and }8\text{ L water}
🟠 EXAMPLE 26: RATIO OF MARKS QUESTION: A student scored 72 marks in Mathematics and 90 marks in Science. Find the ratio of Mathematics marks to Science marks. 🟢 SOLUTION:
72:90 72:90
The HCF is:
18 18
Therefore:
7218:9018 \frac{72}{18}:\frac{90}{18}
=4:5 =4:5
✅ ANSWER:
4:5 4:5
🟠 EXAMPLE 27: RATIO OF SPEEDS QUESTION: The speeds of two cars are in the ratio 3:5. If the first car travels at 60 km/h, find the speed of the second car. 🟢 SOLUTION:
35=60x \frac{3}{5}=\frac{60}{x}
Cross multiply:
3x=60×5 3x=60\times5
3x=300 3x=300
x=100 x=100
✅ ANSWER:
100 km/h 100\text{ km/h}
🟠 EXAMPLE 28: RATIO OF AREAS QUESTION: The sides of two squares are in the ratio 2:3. Find the ratio of their areas. 🟢 SOLUTION: Area of a square is proportional to the square of its side. Therefore:
22:32 2^2:3^2
=4:9 =4:9
✅ ANSWER:
4:9 4:9
🟠 EXAMPLE 29: RATIO OF VOLUMES QUESTION: The radii of two spheres are in the ratio 2:3. Find the ratio of their volumes. 🟢 SOLUTION: Volume of a sphere is proportional to the cube of its radius. Therefore:
23:33 2^3:3^3
=8:27 =8:27
✅ ANSWER:
8:27 8:27
🟠 EXAMPLE 30: RATIO AFTER MULTIPLICATION QUESTION: Two numbers are in the ratio 3:4. If both numbers are multiplied by 5, find the new ratio. 🟢 SOLUTION: Let the numbers be:
3x and 4x 3x\text{ and }4x
After multiplying both by 5:
15x and 20x 15x\text{ and }20x
Therefore:
15x:20x 15x:20x
Cancel x:
15:20 15:20
Simplifying:
3:4 3:4
✅ ANSWER:
3:4 3:4