Recognising direct and inverse proportion
One crate holds 4 bottles. 2 crates hold 8 bottles and 3 crates hold 12 bottles; when the crates increase, the bottles increase in the same proportion. But if 24 chocolates are shared among more children, each child gets fewer chocolates.
When two quantities increase together or decrease together in the same proportion, they are in direct proportion; their ratio (the quotient) does not change. When one quantity increases while the other decreases by the same factor, the pair is inversely proportional; multiplying them always gives the same number.
In the figure: 1 crate holds 4 bottles, 2 crates hold 8, 3 crates hold 12 and 4 crates hold 16 bottles. The ratios 1 : 4, 2 : 8, 3 : 12 and 4 : 16 are all the same.
Rule 1: To spot direct proportion, check the ratio — in every pair, divide the second quantity by the first. If the answer is the same for all pairs, the proportion is direct.
- Easy: 3 notebooks cost 45 rupees and 5 notebooks cost 75 rupees. 45 ÷ 3 = 15 and 75 ÷ 5 = 15, so this is direct proportion.
- Medium: 4 kg of rice costs 280 rupees and 6 kg costs 420 rupees. 280 ÷ 4 = 70 and 420 ÷ 6 = 70, so it is direct proportion.
- Exam type: A motorcycle covers 90 km on 2 litres and 225 km on 5 litres. 90 ÷ 2 = 45 and 225 ÷ 5 = 45, so it is direct proportion.
In the figure: 24 chocolates shared among 2 children give 12 each, among 3 children 8 each, among 4 children 6 each and among 6 children 4 each. 2 × 12, 3 × 8, 4 × 6 and 6 × 4 are all 24.
Rule 2: To spot inverse proportion, check the product — in every pair, multiply the two quantities. If the product is the same for all pairs, the proportion is inverse.
- Easy: 36 pencils shared among 4 children give 9 each, and among 6 children give 6 each. 4 × 9 = 36 and 6 × 6 = 36, so it is inverse proportion.
- Medium: A distance of 180 km takes 6 hours at 30 km/h and 4 hours at 45 km/h. 30 × 6 = 180 and 45 × 4 = 180, so it is inverse proportion.
- Exam type: 8 taps fill a tank in 9 hours and 12 taps fill it in 6 hours. 8 × 9 = 72 and 12 × 6 = 72, so it is inverse proportion.
Rule 3: Look at the direction first, then confirm with arithmetic — if both quantities change in the same direction, they may be in direct proportion; if they change in opposite directions, they may be in inverse proportion. Decide only after checking the ratio or the product.
- Easy: If the number of lamps is doubled, the electricity bill doubles; both move the same way, so it is direct proportion.
- Medium: If more workers are employed, the job finishes in fewer days; opposite directions, so it is inverse proportion.
- Exam type: 5 bags weigh 40 kg and 9 bags weigh 72 kg. 40 ÷ 5 = 8 and 72 ÷ 9 = 8, so the direction and the arithmetic both show direct proportion.
Recognising direct and inverse proportion
In the app you can move this figure with your finger.
Open in the appIn direct proportion the ratio is fixed, and in inverse proportion the product is fixed. In the figure, the red straight line shows direct proportion and the blue curve shows inverse proportion.
Deciding "both increase, so it is direct proportion". Check with the ratio whether the increase is in the same proportion; a child's age and height both increase, but not in the same proportion.
Double one quantity and see. If the other also doubles, it is direct proportion; if it is halved, it is inverse proportion.
Spot the difference
| Point | Direct proportion | Inverse proportion |
|---|---|---|
| How the quantities change | In the same direction | In opposite directions |
| What stays fixed | The ratio (quotient) | The product |
| If one is doubled | The other doubles | The other is halved |
| Example | Articles and their cost | Workers and days of work |
Practice questions
- 17 pens cost 56 rupees and 9 pens cost 72 rupees. Is this direct proportion? Check.
- 230 chocolates shared among 6 children give 5 each, and shared among 10 children give 3 each. Which proportion is this?
- 3Write which of these is direct and which is inverse: (a) the number of articles and the total cost (b) the speed of a vehicle and the time to cover the same distance.
- 4Say whether each set of pairs is in direct proportion, in inverse proportion or neither: (a) (2, 18), (3, 27), (5, 45) (b) (2, 30), (3, 20), (5, 12) (c) (1, 5), (2, 8), (3, 9).
- 5A job takes 12 workers 15 days and takes 20 workers 9 days. Which proportion is this? Show it with arithmetic.
- 6One quantity became 3 times as large while the other changed from 12 to 36. At another time, one quantity became 3 times as large while the other changed from 12 to 4. Which proportion is it each time?
- 7The rice in 5 bags weighs 40 kg. How many kg of rice are in 8 bags? State, with the reason, which proportion you will use.
- 8A hall has 36 seats. They can be set in 4 rows of 9, in 6 rows of 6 or in 12 rows of 3. Which proportion is this? If 9 rows are made, how many seats come in each row?
Show answers (try them yourself first)
- 156 ÷ 7 = 8 and 72 ÷ 9 = 8; the ratios are equal, so it is direct proportion. 2) 6 × 5 = 30 and 10 × 3 = 30; inverse proportion. 3) (a) direct (b) inverse.
- 4(a) 18 ÷ 2 = 9, 27 ÷ 3 = 9, 45 ÷ 5 = 9; direct. (b) 2 × 30 = 60, 3 × 20 = 60, 5 × 12 = 60; inverse. (c) 5 ÷ 1 = 5, 8 ÷ 2 = 4; products 1 × 5 = 5, 2 × 8 = 16; neither.
- 512 × 15 = 180 and 20 × 9 = 180; the product is the same, so it is inverse proportion. 6) The first time it is direct proportion (both became 3 times); the second time it is inverse proportion (the other quantity is 12 ÷ 3 = 4).
- 7Direct proportion, because when the bags increase the weight increases in the same proportion. 40 ÷ 5 = 8 kg in one bag, so 8 × 8 = 64 kg.
- 84 × 9 = 36, 6 × 6 = 36 and 12 × 3 = 36; inverse proportion. With 9 rows there are 36 ÷ 9 = 4 seats in each row.
Finding the unknown in direct proportion
A sweet-maker quotes 1200 rupees for 4 kg of pedas. How much will 7 kg cost? It is easy: first find the cost of 1 kg, then multiply it by 7.
In direct proportion the ratio is fixed. So from a given pair, the unknown can be found in two ways: the unit method (first find the value for one) and the ratio method (write the ratios of the two pairs as equal).
In the figure: 5 articles cost 30 rupees. 30 ÷ 5 = 6, so 1 article costs 6 rupees. Then 8 articles cost 8 × 6 = 48 rupees.
In the figure: With 0, 5, 10 and 15 pencils the rupees are 0, 30, 60 and 90. 5 pencils cost 30 rupees, 10 pencils cost 60 rupees and 15 pencils cost 90 rupees; every pair has the ratio 5 : 30 = 1 : 6.
Rule 1: Unit method — first find the cost of one — divide the given total cost by the number of articles, then multiply by the number you want.
- Easy: 4 kg of pedas cost 1200 rupees. 1 kg costs 1200 ÷ 4 = 300 rupees, so 7 kg cost 7 × 300 = 2100 rupees.
- Medium: 9 metres of cloth cost 1170 rupees. 1 metre costs 1170 ÷ 9 = 130 rupees, so 14 metres cost 14 × 130 = 1820 rupees.
- Exam type: A machine makes 360 items in 5 hours. In 1 hour it makes 360 ÷ 5 = 72 items, so in 8 hours it makes 8 × 72 = 576 items.
Rule 2: Ratio method — write two equal ratios — write the ratio of the first pair equal to the ratio of the second pair and cross-multiply. Keep the order of the quantities the same in both pairs.
- Easy: 2 kg of grapes cost 190 rupees, so 5 kg cost x rupees. 2 : 190 = 5 : x ⇒ 2x = 190 × 5 = 950 ⇒ x = 950 ÷ 2 = 475 rupees.
- Medium: 7 litres of petrol give 84 km, so 11 litres give x km. 7 : 84 = 11 : x ⇒ 7x = 84 × 11 = 924 ⇒ x = 924 ÷ 7 = 132 km.
- Exam type: On a map 3 cm stands for 45 km, so 8 cm stands for x km. 3 : 45 = 8 : x ⇒ 3x = 45 × 8 = 360 ⇒ x = 360 ÷ 3 = 120 km.
Rule 3: Check the answer — find the ratio of both pairs with your answer. If the ratios are equal, the answer is right.
- Easy: 2 kg for 190 rupees and 5 kg for 475 rupees. 190 ÷ 2 = 95 and 475 ÷ 5 = 95, so the answer is right.
- Medium: 7 litres for 84 km and 11 litres for 132 km. 84 ÷ 7 = 12 and 132 ÷ 11 = 12, so the answer is right.
- Exam type: 3 cm for 45 km and 8 cm for 120 km. 45 ÷ 3 = 15 and 120 ÷ 8 = 15, so the answer is right.
Rule 4: Make the units the same first — if one pair has hours and the other minutes, write both in the same unit first, then do the arithmetic.
- Easy: 250 g of ghee costs 150 rupees. 1 kg is 1000 g; 1000 ÷ 250 = 4 times, so the cost is 4 × 150 = 600 rupees.
- Medium: 240 rupees are earned in 2 hours. 2 hours are 120 minutes, so 1 minute earns 240 ÷ 120 = 2 rupees and 45 minutes earn 45 × 2 = 90 rupees.
- Exam type: 3 metres of ribbon cost 90 rupees. 3 metres are 300 cm; 300 : 90 = 80 : x ⇒ 300x = 90 × 80 = 7200 ⇒ x = 7200 ÷ 300 = 24 rupees.
Finding the unknown in direct proportion
In the app you can move this figure with your finger.
Open in the appIn direct proportion, when the articles increase, the cost also increases. Check that your answer is larger than the first cost.
Writing the quantities in the reverse order in the second ratio. If the first is articles : rupees, keep articles : rupees in the second as well.
If the number you want is twice or three times the first number, take the cost twice or three times as well, without dividing.
Spot the difference
| Point | Ratio method | Unit method |
|---|---|---|
| Start | Write two equal ratios | Find the cost of one |
| Operations | Cross-multiplication | Division, then multiplication |
| When easy | When the numbers are awkward | When the cost of one is a whole number |
| Check | Both ratios are equal | Cost of one × number = total |
Practice questions
- 16 pens are bought for 72 rupees. Find the price of 1 pen.
- 2Find the price of 9 notebooks if 4 notebooks are sold for 60 rupees.
- 3Sugar is sold at 225 rupees for 5 kg. Find the price of 8 kg.
- 48 metres of cloth cost 640 rupees. Use the ratio method to find the cost of 13 metres.
- 59 litres of milk cost 405 rupees. Use the unit method to find the cost of 14 litres.
- 6A shopkeeper sells 750 g of cashew nuts for 120 rupees. Find the price of 2 kg at that rate.
- 7A car goes 156 km on 12 litres of petrol. How far will it go on 20 litres? How many litres are needed to go 325 km?
- 8135 rupees are earned in 3 hours of work. How much will be earned in 100 minutes at the same rate?
Show answers (try them yourself first)
- 172 ÷ 6 = 12 rupees. 2) 1 notebook costs 60 ÷ 4 = 15 rupees; 9 × 15 = 135 rupees. 3) 1 kg costs 225 ÷ 5 = 45 rupees; 8 × 45 = 360 rupees.
- 48 : 640 = 13 : x ⇒ 8x = 640 × 13 = 8320 ⇒ x = 8320 ÷ 8 = 1040 rupees.
- 51 litre costs 405 ÷ 9 = 45 rupees; 14 × 45 = 630 rupees. 6) 2 kg is 2000 g; 750 : 120 = 2000 : x ⇒ 750x = 120 × 2000 = 240000 ⇒ x = 240000 ÷ 750 = 320 rupees.
- 71 litre gives 156 ÷ 12 = 13 km; 20 litres give 20 × 13 = 260 km. For 325 km it needs 325 ÷ 13 = 25 litres.
- 83 hours are 180 minutes; 180 : 135 = 100 : x ⇒ 180x = 135 × 100 = 13500 ⇒ x = 13500 ÷ 180 = 75 rupees.
Finding the unknown in inverse proportion
4 girls draw a rangoli in 6 hours. With 12 girls the work takes less time. The total work is 4 × 6 = 24 girl-hours, and 24 ÷ 12 = 2 hours.
When two quantities are inversely proportional, multiplying them gives one fixed number. So multiply the first pair and divide that number by the known value in the second pair.
In the figure: 6 workers work 10 days, 12 workers work 5 days and 15 workers work 4 days. 6 × 10, 12 × 5 and 15 × 4 are all 60, so the three rectangles have the same area.
Rule 1: The product is fixed — multiply first, then divide — find the product of the first pair and divide it by the known number of the second pair.
- Easy: 8 workers do a job in 9 days. 8 × 9 = 72; 12 workers need 72 ÷ 12 = 6 days.
- Medium: 18 workers need 14 days. 18 × 14 = 252; 21 workers need 252 ÷ 21 = 12 days.
- Exam type: 14 workers need 15 days. 14 × 15 = 210; 10 workers need 210 ÷ 10 = 21 days.
Rule 2: Unit method — first find what one needs — find by multiplying how long one person or one tap would take, then divide by the number you want.
- Easy: 6 girls do a job in 4 hours. 1 girl would take 6 × 4 = 24 hours; 8 girls take 24 ÷ 8 = 3 hours.
- Medium: 5 taps fill a tank in 12 hours. 1 tap would take 5 × 12 = 60 hours; 15 taps take 60 ÷ 15 = 4 hours.
- Exam type: 9 workers do a job in 20 days. 1 worker would take 9 × 20 = 180 days; 12 workers take 180 ÷ 12 = 15 days.
Rule 3: For the same distance, speed and time are in inverse proportion — distance = speed × time. The distance stays the same, so when the speed increases, the time decreases in the same proportion.
- Easy: A distance of 240 km takes 4 hours at 60 km/h. At 80 km/h it takes 240 ÷ 80 = 3 hours.
- Medium: A distance of 150 km takes 150 ÷ 25 = 6 hours at 25 km/h and 150 ÷ 50 = 3 hours at 50 km/h.
- Exam type: A train reaches in 3 hours at 54 km/h. The distance is 54 × 3 = 162 km; to reach in 2 hours the speed must be 162 ÷ 2 = 81 km/h.
Rule 4: Make sure the answer is sensible — if the workers increase, the days must decrease. Also the products of both pairs must be equal.
- Easy: 8 workers 9 days and 12 workers 6 days. Workers increased, days decreased; 8 × 9 = 72 and 12 × 6 = 72.
- Medium: 5 taps 12 hours and 15 taps 4 hours. Taps increased, hours decreased; 5 × 12 = 60 and 15 × 4 = 60.
- Exam type: 14 workers 15 days and 10 workers 21 days. Workers decreased, days increased; 14 × 15 = 210 and 10 × 21 = 210.
Finding the unknown in inverse proportion
In the app you can move this figure with your finger.
Open in the appIn inverse proportion you multiply first and then divide. In direct proportion it is the other way round: divide first and then multiply.
Solving inverse proportion the way direct proportion is solved. If 8 workers take 9 days and you find the days for 12 workers as 9 × 12 ÷ 8 = 13.5, the answer is wrong; the right answer is 6 days.
If the number of workers doubles, take half the days; if it triples, take one third of the days.
Spot the difference
| Point | Direct proportion | Inverse proportion |
|---|---|---|
| What is fixed | The ratio | The product |
| Set-up | 3 : 45 = 5 : x | 3 × 45 = 5 × x |
| Finding the value for one | Divide, then multiply | Multiply, then divide |
| How the answer comes | A larger number gives a larger value | A larger number gives a smaller value |
Practice questions
- 16 workers do a job in 20 days. How many days will 10 workers need?
- 24 taps fill a tank in 15 hours. In how many hours will 6 taps fill it?
- 3A distance of 360 km takes 4 hours at 90 km/h. How many hours will it take at 120 km/h?
- 416 workers finish a job in 12 days. How many days will 24 workers need?
- 59 workers do a job in 28 days. In how many days will 12 workers do the same job?
- 65 pumps empty a pond in 8 hours. How many hours will 10 pumps need? How many hours will 4 pumps need?
- 720 workers do a job in 18 days. After 6 days of work, 8 workers leave. In how many days will the remaining workers finish the rest of the job?
- 8A camp has enough grain to last 30 children for 24 days. If 6 more children join, how many days will the same grain last?
Show answers (try them yourself first)
- 16 × 20 = 120; 120 ÷ 10 = 12 days. 2) 4 × 15 = 60; 60 ÷ 6 = 10 hours. 3) 90 × 4 = 360; 360 ÷ 120 = 3 hours.
- 416 × 12 = 192; 192 ÷ 24 = 8 days. 5) 9 × 28 = 252; 252 ÷ 12 = 21 days.
- 65 × 8 = 40; 10 pumps need 40 ÷ 10 = 4 hours and 4 pumps need 40 ÷ 4 = 10 hours.
- 7Total work is 20 × 18 = 360 worker-days. In 6 days 20 × 6 = 120 were done; left 360 − 120 = 240. Remaining workers 20 − 8 = 12; 240 ÷ 12 = 20 days.
- 830 × 24 = 720; children 30 + 6 = 36; 720 ÷ 36 = 20 days.
Direct or inverse proportion in everyday examples
When more passengers board a bus, the total ticket money increases. But if more painters are hired to paint a house, fewer days are needed. Both quantities change, but the relation is different.
In every example ask yourself, "If one quantity doubles, what happens to the other?" If the other also doubles, it is direct proportion; if it is halved, it is inverse proportion; if neither happens, there is no proportion.
In the figure: In direct proportion, when the first quantity goes up (up arrow) the second also goes up, and when the first goes down (down arrow) the second also goes down. In inverse proportion, when the first goes up the second goes down, and when the first goes down the second goes up.
Rule 1: Use the doubling test — double one quantity and see how the other changes.
- Easy: 1 pen costs 10 rupees and 2 pens cost 20 rupees; double the pens, double the cost, so it is direct proportion.
- Medium: 1 tap fills a tank in 12 hours and 2 taps in 6 hours; double the taps, half the time, so it is inverse proportion.
- Exam type: If the speed is doubled, the same distance takes half the time, so speed and time are in inverse proportion.
Rule 2: Check whether the ratio or the product is fixed — work out both the quotients and the products of the given pairs. Decide the proportion by whichever one comes out the same for all pairs.
- Easy: Pairs (3, 45), (5, 75), (8, 120). 45 ÷ 3 = 15, 75 ÷ 5 = 15 and 120 ÷ 8 = 15; the ratio is fixed, so it is direct proportion.
- Medium: Pairs (20, 6), (30, 4), (40, 3). 20 × 6 = 120, 30 × 4 = 120 and 40 × 3 = 120; the product is fixed, so it is inverse proportion.
- Exam type: Pairs (6, 110), (8, 124), (10, 134). The products 6 × 110 = 660, 8 × 124 = 992 and 10 × 134 = 1340 are not equal; the ratios 110 ÷ 6, 124 ÷ 8 and 134 ÷ 10 are not equal either, so there is no proportion.
Direct or inverse proportion in everyday examples
In the app you can move this figure with your finger.
Open in the appRule 3: Not every relation is a proportion — two quantities may change together without being in direct or inverse proportion. If both tests fail, write that there is no proportion.
- Easy: A child's age and height both increase, but the height does not grow in step with the age; so it is not direct proportion.
- Medium: A square of side 2 has area 4 and a square of side 4 has area 16. The side doubled but the area became four times; 4 ÷ 2 = 2 and 16 ÷ 4 = 4 are not equal, so it is not direct proportion.
- Exam type: A taxi fare is 40 rupees for 2 km and 60 rupees for 4 km. 40 ÷ 2 = 20 and 60 ÷ 4 = 15 are not equal, so it is not direct proportion.
"Both increase" alone does not make it direct proportion. The increase must be in the same proportion.
Thinking "more children, more chocolates". When the total chocolates are the same, more children means fewer for each, so it is inverse proportion.
If the total stays the same (distance, work, total chocolates), it is inverse proportion. If the rate stays the same (cost of one article), it is direct proportion.
Spot the difference
| Situation | Relation | What stays fixed |
|---|---|---|
| Articles and their cost | Direct | Cost of one article (ratio) |
| Workers and days of work | Inverse | Total work (product) |
| Speed and time (same distance) | Inverse | Distance (product) |
| Petrol and distance covered | Direct | Distance on 1 litre (ratio) |
Practice questions
- 1Write direct or inverse: (a) the speed of a cycle and the time to go 24 km (b) the number of dozens and the total cost of eggs (c) the days needed to read 60 pages and the pages read per day.
- 2Which proportion are the pairs (4, 28), (6, 42), (9, 63) in?
- 3Which proportion are the pairs (3, 40), (5, 24), (8, 15) in?
- 4Are the pairs (2, 9), (4, 16), (6, 25) in direct proportion, in inverse proportion or neither? Check.
- 524 children go on a trip and each pays 150 rupees. If the total cost stays the same and 30 children go, how much should each pay?
- 65 classes have 40 benches. If each class is to get the same number of benches, how many benches do 8 classes need?
- 7Keep the area of a rectangle 60 sq cm and take the length as 5, 6 and 10 cm. What will the width be? Which proportion is this?
- 8The values of one quantity are 6, 9, 15 and of another are 4, 6, 10. Which proportion is this?
Show answers (try them yourself first)
- 1(a) inverse (b) direct (c) inverse. 2) 28 ÷ 4 = 7, 42 ÷ 6 = 7 and 63 ÷ 9 = 7; direct proportion. 3) 3 × 40 = 120, 5 × 24 = 120 and 8 × 15 = 120; inverse proportion.
- 49 ÷ 2 = 4.5 and 16 ÷ 4 = 4; also 2 × 9 = 18 and 4 × 16 = 64; neither is the same, so neither.
- 5Total cost 24 × 150 = 3600 rupees; 3600 ÷ 30 = 120 rupees each (inverse proportion).
- 61 class gets 40 ÷ 5 = 8 benches; 8 classes need 8 × 8 = 64 benches (direct proportion).
- 7The width is 60 ÷ 5 = 12, 60 ÷ 6 = 10 and 60 ÷ 10 = 6 cm; length × width is fixed, so it is inverse proportion.
- 86 ÷ 4 = 1.5, 9 ÷ 6 = 1.5 and 15 ÷ 10 = 1.5; the ratio 6 : 4 = 9 : 6 = 15 : 10 = 3 : 2; direct proportion.
Partnership
Two friends start a snack stall together. One puts in 2000 rupees and the other 3000 rupees. If the profit is 1500 rupees, sharing it equally is not fair; the one who invested more gets a bigger share.
People who put money together into a business are called partners and their money is called the capital. The profit or loss is shared among the partners in the ratio of their capitals.
In the figure: The capitals 2000 and 3000 are in the ratio 2 : 3. The sum of the parts is 2 + 3 = 5; the profit 1500 ÷ 5 = 300 is one part. The shares are 2 × 300 = 600 and 3 × 300 = 900.
Rule 1: Simplify the ratio of the capitals — divide all the capitals by their common divisor and write the ratio in small numbers.
- Easy: Capitals 4000 and 6000. Dividing both by 2000 gives 4000 : 6000 = 2 : 3.
- Medium: Three partners with capitals 3000, 4500 and 7500. Dividing by 1500 gives 3000 : 4500 : 7500 = 2 : 3 : 5.
- Exam type: Capitals 7000 and 3500. Dividing by 3500 gives 7000 : 3500 = 2 : 1.
Rule 2: Parts method — sharing a profit between two partners — add the parts, divide the profit by that sum to get one part, and multiply each partner's parts by it.
- Easy: Ratio 1 : 2, profit 1800 rupees. Parts 1 + 2 = 3; one part 1800 ÷ 3 = 600; shares 600 and 2 × 600 = 1200 rupees.
- Medium: Ratio 3 : 4, profit 4200 rupees. Parts 3 + 4 = 7; one part 4200 ÷ 7 = 600; shares 3 × 600 = 1800 and 4 × 600 = 2400 rupees.
- Exam type: Ratio 5 : 7, profit 9600 rupees. Parts 5 + 7 = 12; one part 9600 ÷ 12 = 800; shares 5 × 800 = 4000 and 7 × 800 = 5600 rupees.
In the figure: Step 1: capitals 1200 : 1800 : 3000 = 2 : 3 : 5. Step 2: sum of the parts 2 + 3 + 5 = 10. Step 3: one part 5000 ÷ 10 = 500. Step 4: shares 2 × 500 = 1000, 3 × 500 = 1500 and 5 × 500 = 2500.
Rule 3: Three partners — same method, three parts — write the ratio of all three capitals, add the three parts, find one part and then each partner's share.
- Easy: Ratio 1 : 1 : 2, profit 8000 rupees. Parts 1 + 1 + 2 = 4; one part 8000 ÷ 4 = 2000; shares 2000, 2000 and 4000 rupees.
- Medium: Ratio 2 : 3 : 4, profit 6300 rupees. Parts 2 + 3 + 4 = 9; one part 6300 ÷ 9 = 700; shares 1400, 2100 and 2800 rupees.
- Exam type: Capitals 1500, 2500 and 3500, so the ratio is 3 : 5 : 7. Profit 9000 rupees. Parts 3 + 5 + 7 = 15; one part 9000 ÷ 15 = 600; shares 1800, 3000 and 4200 rupees.
Rule 4: If the profit or loss is given in per cent, find the amount first — find the given per cent of the total capital and then share that amount in the ratio.
- Easy: Total capital 40000 rupees and profit 10%. The profit is 40000 × 10 ÷ 100 = 4000 rupees.
- Medium: Capitals 25000 and 35000, total 60000. A 5% loss is 60000 × 5 ÷ 100 = 3000 rupees. Ratio 5 : 7, parts 12; one part 3000 ÷ 12 = 250; the loss shares are 1250 and 1750 rupees.
- Exam type: Capitals 30000 and 50000, total 80000. A 15% profit is 80000 × 15 ÷ 100 = 12000 rupees. Ratio 3 : 5, parts 8; one part 12000 ÷ 8 = 1500; shares 4500 and 7500 rupees.
Partnership
In the app you can move this figure with your finger.
Open in the appBoth profit and loss are shared in the ratio of the capitals. All the shares together give the total profit or loss again.
Sharing the profit equally among the partners, or working with big numbers without simplifying the ratio and making errors.
First find how many rupees one part is. Then each partner's share is his parts × one part. At the end, check that all the shares add up to the profit.
Spot the difference
| Step | What to do | Example (capitals 2000 and 3000, profit 1500) |
|---|---|---|
| Step 1 | Simplify the ratio | 2000 : 3000 = 2 : 3 |
| Step 2 | Add the parts | 2 + 3 = 5 |
| Step 3 | Find one part | 1500 ÷ 5 = 300 |
| Step 4 | Find each share | 600 and 900 |
Practice questions
- 1Two partners have capitals of 3000 and 5000 rupees. Write their ratio in simplest form.
- 2The partners' ratio is 2 : 3 and the profit is 2500 rupees. What is each share?
- 3The partners' ratio is 1 : 4 and the profit is 3500 rupees. What is each share?
- 4The capitals are 6000 and 9000 rupees and the profit is 4000 rupees. Find each share.
- 5The partners' ratio is 4 : 5 and the profit is 8100 rupees. What is each share?
- 6Three partners have capitals of 3000, 4000 and 5000 rupees. If the profit is 7200 rupees, what is each share?
- 7A invested 20000 and B invested 30000 rupees in a business and there was a loss of 12% of the capital. How much of the loss does each partner bear?
- 8The capitals of A and B are in the ratio 3 : 4. If B's share of the profit is 3000 rupees, what is the total profit and what is A's share?
Show answers (try them yourself first)
- 13000 : 5000 = 3 : 5. 2) Parts 2 + 3 = 5; one part 2500 ÷ 5 = 500; shares 1000 and 1500 rupees. 3) Parts 1 + 4 = 5; one part 3500 ÷ 5 = 700; shares 700 and 2800 rupees.
- 46000 : 9000 = 2 : 3; parts 5; one part 4000 ÷ 5 = 800; shares 1600 and 2400 rupees.
- 5Parts 4 + 5 = 9; one part 8100 ÷ 9 = 900; shares 3600 and 4500 rupees.
- 63000 : 4000 : 5000 = 3 : 4 : 5; parts 12; one part 7200 ÷ 12 = 600; shares 1800, 2400 and 3000 rupees.
- 7Total capital 20000 + 30000 = 50000; loss 50000 × 12 ÷ 100 = 6000 rupees. Ratio 2 : 3, parts 5; one part 6000 ÷ 5 = 1200; losses 2400 and 3600 rupees.
- 8B's 4 parts are 3000 rupees, so one part is 3000 ÷ 4 = 750. A's share is 3 × 750 = 2250 rupees and the total profit is 7 × 750 = 5250 rupees.