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Quantitative Aptitude · Ratio and Proportion

Proportion - Formulas, Key Points and Examples

Explanation

🔵 PROPORTION 🟢 1. BASIC CONCEPT OF PROPORTION A proportion is a statement that two ratios are equal. If:
a:b=c:d a:b=c:d
then:
ab=cd \frac{a}{b}=\frac{c}{d}
This is called a proportion. The terms a and d are called the extremes. The terms b and c are called the means. 🟢 2. FUNDAMENTAL RULE OF PROPORTION If:
a:b=c:d a:b=c:d
then, by cross multiplication:
ad=bc ad=bc
This is the most important rule used for solving proportion problems. 🟢 3. FINDING AN UNKNOWN TERM If:
a:b=c:x a:b=c:x
then:
ax=bc ax=bc
Therefore:
x=bca x=\frac{bc}{a}
🟢 4. FOURTH PROPORTIONAL If:
a:b=c:x a:b=c:x
then x is called the fourth proportional to a, b and c. Formula:
x=bca x=\frac{bc}{a}
🟢 5. THIRD PROPORTIONAL If:
a:b=b:x a:b=b:x
then x is called the third proportional to a and b. Using cross multiplication:
ax=b2 ax=b^2
Therefore:
x=b2a x=\frac{b^2}{a}
🟢 6. MEAN PROPORTIONAL If x is the mean proportional between a and b:
a:x=x:b a:x=x:b
Then:
x2=ab x^2=ab
Therefore:
x=ab x=\sqrt{ab}
🟢 7. 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 relationship is:
b2=ac b^2=ac
🟢 8. DIRECT PROPORTION Two quantities are in direct proportion when they increase or decrease together in the same ratio. For example, if the number of items increases, the total cost increases proportionally when the price per item remains constant. We write:
x∝y x\propto y
Therefore:
xy=k \frac{x}{y}=k
where k is a constant. For two situations:
x1y1=x2y2 \frac{x_1}{y_1} = \frac{x_2}{y_2}
🟢 9. DIRECT PROPORTION SHORTCUT If x and y are directly proportional:
x1:y1=x2:y2 x_1:y_1=x_2:y_2
Therefore:
x1y2=x2y1 x_1y_2=x_2y_1
🟢 10. INVERSE PROPORTION Two quantities are in inverse proportion when an increase in one quantity causes a proportional decrease in the other. For example, when the number of workers increases, the number of days required to complete the same work decreases. We write:
x∝1y x\propto\frac{1}{y}
Therefore:
xy=k xy=k
🟢 11. INVERSE PROPORTION SHORTCUT If x and y are inversely proportional:
x1y1=x2y2 x_1y_1=x_2y_2
🟢 12. DIRECT VS INVERSE PROPORTION Direct proportion:
xy=k \frac{x}{y}=k
Inverse proportion:
xy=k xy=k
🟢 13. PROPORTION WITH THREE QUANTITIES If:
a:b:c=x:y:z a:b:c=x:y:z
then corresponding terms can be compared separately. For example:
ax=by=cz \frac{a}{x} = \frac{b}{y} = \frac{c}{z}
🟢 14. COMPOUND PROPORTION Compound proportion involves three or more quantities where the required quantity depends on more than one factor. For example, the number of workers, number of days and number of hours per day may all affect the amount of work completed. First determine whether each quantity has a direct or inverse relationship with the required quantity. 🟢 15. WORK AND PROPORTION For the same amount of work: More workers → fewer days. Fewer workers → more days. Therefore:
Workers×Days=Constant \text{Workers}\times\text{Days} = \text{Constant}
🟢 16. WORKERS AND HOURS For the same work:
Workers×Days×Hours per day=Constant \text{Workers}\times\text{Days}\times\text{Hours per day} = \text{Constant}
This is useful in compound proportion problems. 🟢 17. SPEED AND TIME For a fixed distance: Higher speed → less time. Lower speed → more time. Therefore:
Speed×Time=Constant \text{Speed}\times\text{Time} = \text{Constant}
🟢 18. DISTANCE AND TIME For constant speed:
DistanceTime=Constant \frac{\text{Distance}}{\text{Time}} = \text{Constant}
Therefore:
D1T1=D2T2 \frac{D_1}{T_1} = \frac{D_2}{T_2}
🟢 19. COST AND QUANTITY If the price per item remains constant:
Cost∝Quantity \text{Cost}\propto\text{Quantity}
Therefore:
C1Q1=C2Q2 \frac{C_1}{Q_1} = \frac{C_2}{Q_2}
🟢 20. RATIO AND PROPORTION A ratio compares two quantities. A proportion states that two ratios are equal. Ratio:
a:b a:b
Proportion:
a:b=c:d a:b=c:d
🟢 21. IMPORTANT PROPERTIES OF PROPORTION If:
ab=cd \frac{a}{b} = \frac{c}{d}
then:
ad=bc ad=bc
Also:
ba=dc \frac{b}{a} = \frac{d}{c}
and:
ac=bd \frac{a}{c} = \frac{b}{d}
🟢 22. ADDENDO PROPERTY If:
ab=cd \frac{a}{b} = \frac{c}{d}
then:
a+bb=c+dd \frac{a+b}{b} = \frac{c+d}{d}
🟢 23. SUBTRACTENDO PROPERTY If:
ab=cd \frac{a}{b} = \frac{c}{d}
then:
a−bb=c−dd \frac{a-b}{b} = \frac{c-d}{d}
🟢 24. DIVIDENDO PROPERTY If:
ab=cd \frac{a}{b} = \frac{c}{d}
then:
ab=cd \frac{a}{b} = \frac{c}{d}
and the corresponding divided forms can be obtained by dividing the numerator and denominator by the same non-zero quantity. 🟢 25. COMBINED ADDITION AND SUBTRACTION If:
ab=cd \frac{a}{b} = \frac{c}{d}
then:
a+ba−b=c+dc−d \frac{a+b}{a-b} = \frac{c+d}{c-d}
provided the denominators are non-zero. 🟡 KEY POINTS 1. Proportion means equality of two ratios. 2. The fastest method for most basic proportion problems is cross multiplication. 3. In:
a:b=c:d a:b=c:d
the product of extremes equals the product of means:
ad=bc ad=bc
4. Direct proportion:
xy=k \frac{x}{y}=k
5. Inverse proportion:
xy=k xy=k
6. For direct proportion, both quantities move in the same direction. 7. For inverse proportion, the quantities move in opposite directions. 8. Before solving a word problem, identify whether the relationship is direct or inverse. 9. For the same work:
Workers×Days=Constant \text{Workers}\times\text{Days} = \text{Constant}
10. For the same distance:
Speed×Time=Constant \text{Speed}\times\text{Time} = \text{Constant}
11. Fourth proportional:
x=bca x=\frac{bc}{a}
12. Third proportional:
x=b2a x=\frac{b^2}{a}
13. Mean proportional:
x=ab x=\sqrt{ab}
14. Continued proportion:
b2=ac b^2=ac
🔴 EXAM TIP First identify the type of relationship. Direct → use equal ratios. Inverse → use equal products. Then form the proportion and cross multiply.

Example

🟠 EXAMPLE 1: BASIC PROPORTION QUESTION: Check whether 3:5 and 12:20 are in proportion. 🟢 SOLUTION: Write the ratios:
35 \frac{3}{5}
and:
1220 \frac{12}{20}
Cross multiply:
3×20=60 3\times20=60
5×12=60 5\times12=60
Since both products are equal, the ratios are in proportion. ✅ ANSWER:
Yes, they are in proportion. \text{Yes, they are in proportion.}
🟠 EXAMPLE 2: FINDING AN UNKNOWN TERM QUESTION: If 4:7 = 12:x, find x. 🟢 SOLUTION:
47=12x \frac{4}{7} = \frac{12}{x}
Cross multiply:
4x=7×12 4x=7\times12
4x=84 4x=84
x=21 x=21
✅ ANSWER:
21 21
🟠 EXAMPLE 3: FOURTH PROPORTIONAL QUESTION: Find the fourth proportional to 3, 5 and 15. 🟢 SOLUTION: Let the fourth proportional be x.
3:5=15:x 3:5=15:x
Cross multiply:
3x=5×15 3x=5\times15
3x=75 3x=75
x=25 x=25
✅ ANSWER:
25 25
🟠 EXAMPLE 4: THIRD PROPORTIONAL QUESTION: Find the third proportional to 4 and 8. 🟢 SOLUTION: Let the third proportional be x.
4:8=8:x 4:8=8:x
Cross multiply:
4x=8×8 4x=8\times8
4x=64 4x=64
x=16 x=16
✅ ANSWER:
16 16
🟠 EXAMPLE 5: MEAN PROPORTIONAL QUESTION: Find the mean proportional between 9 and 16. 🟢 SOLUTION:
x=9×16 x=\sqrt{9\times16}
=144 =\sqrt{144}
=12 =12
✅ ANSWER:
12 12
🟠 EXAMPLE 6: CONTINUED PROPORTION QUESTION: If 2, 6 and x are in continued proportion, find x. 🟢 SOLUTION:
2:6=6:x 2:6=6:x
Cross multiply:
2x=36 2x=36
x=18 x=18
✅ ANSWER:
18 18
🟠 EXAMPLE 7: DIRECT PROPORTION QUESTION: If 5 notebooks cost ₹250, find the cost of 8 notebooks. 🟢 SOLUTION: Cost and number of notebooks are directly proportional.
2505=x8 \frac{250}{5} = \frac{x}{8}
Cross multiply:
5x=250×8 5x=250\times8
5x=2000 5x=2000
x=400 x=400
✅ ANSWER:
₹400 ₹400
🟠 EXAMPLE 8: DIRECT PROPORTION QUESTION: A car travels 180 km in 3 hours at constant speed. How far will it travel in 5 hours? 🟢 SOLUTION: Distance and time are directly proportional.
1803=x5 \frac{180}{3} = \frac{x}{5}
Cross multiply:
3x=180×5 3x=180\times5
3x=900 3x=900
x=300 x=300
✅ ANSWER:
300 km 300\text{ km}
🟠 EXAMPLE 9: INVERSE PROPORTION QUESTION: 6 workers can complete a work in 12 days. How many days will 8 workers take to complete the same work? 🟢 SOLUTION: Workers and days are inversely proportional. Therefore:
6×12=8×x 6\times12=8\times x
72=8x 72=8x
x=9 x=9
✅ ANSWER:
9 days 9\text{ days}
🟠 EXAMPLE 10: INVERSE PROPORTION QUESTION: A journey takes 6 hours at a speed of 50 km/h. How long will it take at 75 km/h for the same distance? 🟢 SOLUTION: Speed and time are inversely proportional.
50×6=75×x 50\times6=75\times x
300=75x 300=75x
x=4 x=4
✅ ANSWER:
4 hours 4\text{ hours}
🟠 EXAMPLE 11: WORKERS AND DAYS QUESTION: 10 workers can complete a job in 18 days. How many workers are required to complete the same job in 12 days? 🟢 SOLUTION: Workers and days are inversely proportional.
10×18=x×12 10\times18=x\times12
180=12x 180=12x
x=15 x=15
✅ ANSWER:
15 workers 15\text{ workers}
🟠 EXAMPLE 12: WORKERS AND HOURS QUESTION: 8 workers working 6 hours per day can complete a job in 15 days. How many days will 12 workers take if they work 5 hours per day? 🟢 SOLUTION: For the same work:
Workers×Hours per day×Days=Constant \text{Workers}\times\text{Hours per day}\times\text{Days} = \text{Constant}
Therefore:
8×6×15=12×5×x 8\times6\times15 = 12\times5\times x
720=60x 720=60x
x=12 x=12
✅ ANSWER:
12 days 12\text{ days}
🟠 EXAMPLE 13: DIRECT PROPORTION WITH COST QUESTION: If 12 kg of rice costs ₹720, what will 20 kg cost? 🟢 SOLUTION: Cost and quantity are directly proportional.
72012=x20 \frac{720}{12} = \frac{x}{20}
Cross multiply:
12x=720×20 12x=720\times20
12x=14400 12x=14400
x=1200 x=1200
✅ ANSWER:
₹1200 ₹1200
🟠 EXAMPLE 14: INVERSE PROPORTION WITH WORKERS QUESTION: 15 workers can complete a work in 20 days. How many days will 25 workers take? 🟢 SOLUTION: Workers and days are inversely proportional.
15×20=25×x 15\times20=25\times x
300=25x 300=25x
x=12 x=12
✅ ANSWER:
12 days 12\text{ days}
🟠 EXAMPLE 15: FINDING A MISSING TERM QUESTION: If 7:9 = x:27, find x. 🟢 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 16: DIRECT PROPORTION WITH DISTANCE QUESTION: A train travels 240 km in 4 hours. How far will it travel in 7 hours at the same speed? 🟢 SOLUTION: Distance and time are directly proportional.
2404=x7 \frac{240}{4} = \frac{x}{7}
4x=240×7 4x=240\times7
4x=1680 4x=1680
x=420 x=420
✅ ANSWER:
420 km 420\text{ km}
🟠 EXAMPLE 17: INVERSE PROPORTION WITH SPEED QUESTION: A car takes 8 hours to complete a journey at 60 km/h. How much time will it take at 80 km/h? 🟢 SOLUTION: Speed and time are inversely proportional.
60×8=80×x 60\times8=80\times x
480=80x 480=80x
x=6 x=6
✅ ANSWER:
6 hours 6\text{ hours}
🟠 EXAMPLE 18: COMPOUND PROPORTION QUESTION: 12 workers working 8 hours per day can complete a job in 10 days. How many days will 16 workers working 6 hours per day take to complete the same job? 🟢 SOLUTION: For the same work:
12×8×10=16×6×x 12\times8\times10 = 16\times6\times x
960=96x 960=96x
x=10 x=10
✅ ANSWER:
10 days 10\text{ days}
🟠 EXAMPLE 19: DIRECT PROPORTION WITH PRICE QUESTION: 5 pens cost ₹75. Find the cost of 18 pens. 🟢 SOLUTION: Cost and number of pens are directly proportional.
755=x18 \frac{75}{5} = \frac{x}{18}
5x=75×18 5x=75\times18
5x=1350 5x=1350
x=270 x=270
✅ ANSWER:
₹270 ₹270
🟠 EXAMPLE 20: INVERSE PROPORTION QUESTION: 4 machines can produce a certain quantity of goods in 15 hours. How many hours will 6 machines take to produce the same quantity? 🟢 SOLUTION: Machines and time are inversely proportional.
4×15=6×x 4\times15=6\times x
60=6x 60=6x
x=10 x=10
✅ ANSWER:
10 hours 10\text{ hours}
🟠 EXAMPLE 21: MEAN PROPORTIONAL QUESTION: Find the mean proportional between 12 and 27. 🟢 SOLUTION:
x=12×27 x=\sqrt{12\times27}
=324 =\sqrt{324}
=18 =18
✅ ANSWER:
18 18
🟠 EXAMPLE 22: CONTINUED PROPORTION QUESTION: Find x if 5, 10 and x are in continued proportion. 🟢 SOLUTION:
5:10=10:x 5:10=10:x
Cross multiply:
5x=100 5x=100
x=20 x=20
✅ ANSWER:
20 20
🟠 EXAMPLE 23: DIRECT PROPORTION QUESTION: A worker earns ₹900 for 6 days of work. How much will he earn for 10 days at the same daily rate? 🟢 SOLUTION: Earnings and number of days are directly proportional.
9006=x10 \frac{900}{6} = \frac{x}{10}
6x=900×10 6x=900\times10
6x=9000 6x=9000
x=1500 x=1500
✅ ANSWER:
₹1500 ₹1500
🟠 EXAMPLE 24: INVERSE PROPORTION QUESTION: A stock of food is sufficient for 20 people for 15 days. If only 12 people consume it, for how many days will it last? 🟢 SOLUTION: Number of people and number of days are inversely proportional.
20×15=12×x 20\times15=12\times x
300=12x 300=12x
x=25 x=25
✅ ANSWER:
25 days 25\text{ days}
🟠 EXAMPLE 25: COMPOUND PROPORTION QUESTION: 8 workers working 7 hours per day complete a work in 15 days. How many days will 12 workers working 5 hours per day take to complete the same work? 🟢 SOLUTION: For the same work:
8×7×15=12×5×x 8\times7\times15 = 12\times5\times x
840=60x 840=60x
x=14 x=14
✅ ANSWER:
14 days 14\text{ days}
🟠 EXAMPLE 26: FINDING FOURTH PROPORTIONAL QUESTION: Find the fourth proportional to 4, 6 and 12. 🟢 SOLUTION: Let the fourth proportional be x.
4:6=12:x 4:6=12:x
4x=6×12 4x=6\times12
4x=72 4x=72
x=18 x=18
✅ ANSWER:
18 18
🟠 EXAMPLE 27: DIRECT PROPORTION WITH QUANTITY QUESTION: If 15 metres of cloth costs ₹1,200, find the cost of 22 metres. 🟢 SOLUTION: Cost and quantity are directly proportional.
120015=x22 \frac{1200}{15} = \frac{x}{22}
15x=1200×22 15x=1200\times22
15x=26400 15x=26400
x=1760 x=1760
✅ ANSWER:
₹1760 ₹1760
🟠 EXAMPLE 28: MIXED DIRECT AND INVERSE PROPORTION QUESTION: 6 workers working 8 hours per day can complete a job in 20 days. How many days will 10 workers working 6 hours per day take to complete the same job? 🟢 SOLUTION: For the same work:
6×8×20=10×6×x 6\times8\times20 = 10\times6\times x
960=60x 960=60x
x=16 x=16
✅ ANSWER:
16 days 16\text{ days}