Two Prices Are Only Comparable in the Same Unit
Learn to solve word problems by setting up a proportion, split a quantity in a given ratio using share fractions, convert between units proportionally, and compare two rates fairly.
Which is cheaper, 5 kg for 200 rupees or 3 kg for 126 rupees?
The first. Bring both to the same unit — the price of one kilogram — and the comparison becomes obvious:
Two rates can only be compared once they are expressed the same way. This page covers everything in the CBSE Class 8 Mathematics chapter's second part: proportion word problems, dividing a quantity in a ratio, unit conversion, and comparing rates.
Two rates can only be compared once they are expressed the same way. This page covers everything in the CBSE Class 8 Mathematics chapter's second part: proportion word problems, dividing a quantity in a ratio, unit conversion, and comparing rates.
How do you solve a word problem by setting up a proportion?
Write the two matching quantities as a proportion, keeping like with like, then cross-multiply. Or find the value for one unit first — the unitary method — and scale up.
Worked example. If kg of rice costs ₹150, what do kg cost?
By proportion. Setting up :
By the unitary method. One kilogram costs , so seven cost . The same answer.
Worked example. A car travels km on litres of petrol. How far on litres?
Distance for 1 litre km, so for 20 litres it is km.
Worked example running backwards. If m of cloth costs ₹294, how much cloth for ₹504?
Cost of 1 m , so ₹504 buys m.
Worked example. If 8 workers can pack 96 boxes, how many boxes can 5 workers pack at the same rate?
One worker packs boxes, so 5 workers pack boxes.
A size check protects every answer. More kilograms must cost more, so ₹350 for 7 kg against ₹150 for 3 kg is believable — and an answer smaller than ₹150 would show the multiplication had gone the wrong way.
Worked example. If kg of rice costs ₹150, what do kg cost?
By proportion. Setting up :
By the unitary method. One kilogram costs , so seven cost . The same answer.
Worked example. A car travels km on litres of petrol. How far on litres?
Distance for 1 litre km, so for 20 litres it is km.
Worked example running backwards. If m of cloth costs ₹294, how much cloth for ₹504?
Cost of 1 m , so ₹504 buys m.
Worked example. If 8 workers can pack 96 boxes, how many boxes can 5 workers pack at the same rate?
One worker packs boxes, so 5 workers pack boxes.
A size check protects every answer. More kilograms must cost more, so ₹350 for 7 kg against ₹150 for 3 kg is believable — and an answer smaller than ₹150 would show the multiplication had gone the wrong way.
Formula
How do you divide a quantity in a given ratio?
Use the share fractions. To divide a quantity in the ratio :
The denominator is the total number of parts.
Worked example. Divide ₹1200 in the ratio . The total parts are :
Check: , and . Both conditions hold.
Worked example. Divide ₹2500 in the ratio . Total parts :
Check: .
Three-way division. Divide sweets among three children in the ratio . Total parts :
Check: .
From one share to the whole. If the smaller share of a division is ₹480, the whole is .
The denominator must be the total parts, not the other term of the ratio. Writing for the first share gives ₹800, and the two shares would then add to more than the money available — which is exactly what the sum check catches.
The denominator is the total number of parts.
Worked example. Divide ₹1200 in the ratio . The total parts are :
Check: , and . Both conditions hold.
Worked example. Divide ₹2500 in the ratio . Total parts :
Check: .
Three-way division. Divide sweets among three children in the ratio . Total parts :
Check: .
From one share to the whole. If the smaller share of a division is ₹480, the whole is .
The denominator must be the total parts, not the other term of the ratio. Writing for the first share gives ₹800, and the two shares would then add to more than the money available — which is exactly what the sum check catches.
How do you convert between units using proportional reasoning?
Every conversion is a proportion, because the two units are related by a fixed factor.
The factors to know:
- Length — km m, m cm, cm mm
- Mass — kg g, quintal kg
- Volume — L mL, m³ L
- Currency — ₹1 paise
- Time — hour minutes, minute seconds, day hours
Worked conversions.
Going the other way, divide:
Worked example in context. A tank holds m³ of water. In litres that is L, and filling it with L buckets needs buckets.
The direction is what to settle before calculating. Converting to a smaller unit means multiplying, so the number gets bigger — 2.5 km becomes 2500 m. Converting to a larger unit means dividing. A quick sense check on the size of the answer catches a reversed conversion immediately.
The factors to know:
- Length — km m, m cm, cm mm
- Mass — kg g, quintal kg
- Volume — L mL, m³ L
- Currency — ₹1 paise
- Time — hour minutes, minute seconds, day hours
Worked conversions.
Going the other way, divide:
Worked example in context. A tank holds m³ of water. In litres that is L, and filling it with L buckets needs buckets.
The direction is what to settle before calculating. Converting to a smaller unit means multiplying, so the number gets bigger — 2.5 km becomes 2500 m. Converting to a larger unit means dividing. A quick sense check on the size of the answer catches a reversed conversion immediately.
How do you compare two rates given in different units?
Convert both to a common unit, then compare.
Comparing prices. Shop A sells kg for ₹200; Shop B sells kg for ₹126. Reduce both to the price per kilogram:
Shop A is cheaper, by ₹2 per kg.
Comparing speeds in different units. Is km/h faster or slower than m/s? Convert the first using km/h m/s:
Since , the second is faster. Converting the other way instead: km/h, which is more than 72 — the same conclusion, as it must be.
Comparing fuel efficiency. Car P does km on L, so km per litre. Car Q does km on L, so km per litre. Car P is more efficient.
Comparing work rates. A types words in minutes, so words per minute. B types words in minutes, so per minute. A is faster.
The choice of common unit is free, but it must be the same for both, and the direction of the comparison is worth thinking about. For prices the lower rate per kilogram is better; for efficiency the higher kilometres per litre is better — so decide which way "better" runs before announcing an answer.
Comparing prices. Shop A sells kg for ₹200; Shop B sells kg for ₹126. Reduce both to the price per kilogram:
Shop A is cheaper, by ₹2 per kg.
Comparing speeds in different units. Is km/h faster or slower than m/s? Convert the first using km/h m/s:
Since , the second is faster. Converting the other way instead: km/h, which is more than 72 — the same conclusion, as it must be.
Comparing fuel efficiency. Car P does km on L, so km per litre. Car Q does km on L, so km per litre. Car P is more efficient.
Comparing work rates. A types words in minutes, so words per minute. B types words in minutes, so per minute. A is faster.
The choice of common unit is free, but it must be the same for both, and the direction of the comparison is worth thinking about. For prices the lower rate per kilogram is better; for efficiency the higher kilometres per litre is better — so decide which way "better" runs before announcing an answer.
Exam tip
Exam tip: checking that the shares add back up
Proportional-reasoning questions carry an easy self-check, so use it.
After dividing a quantity in a ratio, add the shares and confirm they give the original total — and confirm they reduce back to the given ratio. Two lines, and the answer is proved.
Use with the total parts on the bottom, never the other term.
For unit conversion, decide the direction first: to a smaller unit means multiply, to a larger unit means divide. Then sense-check the size of the answer.
When comparing rates, convert both to the same unit and say which unit you chose — price per kg, km per litre, words per minute.
And state clearly which is better and why: Shop A is cheaper at ₹40 per kg against ₹42. A bare pair of numbers does not answer the question.
After dividing a quantity in a ratio, add the shares and confirm they give the original total — and confirm they reduce back to the given ratio. Two lines, and the answer is proved.
Use with the total parts on the bottom, never the other term.
For unit conversion, decide the direction first: to a smaller unit means multiply, to a larger unit means divide. Then sense-check the size of the answer.
When comparing rates, convert both to the same unit and say which unit you chose — price per kg, km per litre, words per minute.
And state clearly which is better and why: Shop A is cheaper at ₹40 per kg against ₹42. A bare pair of numbers does not answer the question.
Did you know
Why does the cheaper shop sometimes look dearer at first glance?
Because the totals are not comparable until the quantities match.
₹126 looks less than ₹200, so Shop B seems cheaper — but it is selling less rice. Only when both are reduced to one kilogram does the real comparison appear, and then Shop A wins at ₹40 against ₹42.
That is the whole purpose of a rate: it strips out the quantity so two offers can be judged side by side. It is also why shop shelves print a price per kilogram or per litre in small type beneath the pack price — the pack prices alone are designed to be hard to compare.
₹126 looks less than ₹200, so Shop B seems cheaper — but it is selling less rice. Only when both are reduced to one kilogram does the real comparison appear, and then Shop A wins at ₹40 against ₹42.
That is the whole purpose of a rate: it strips out the quantity so two offers can be judged side by side. It is also why shop shelves print a price per kilogram or per litre in small type beneath the pack price — the pack prices alone are designed to be hard to compare.
Key takeaways
Dividing in a ratio and comparing rates: quick revision
- Solve proportion problems by cross-multiplying, or by the unitary method — find the value for one unit, then scale.
- To divide in the ratio , the shares are and — the denominator is the total parts.
- ₹1200 in gives ₹480 and ₹720, and in gives , and — always check the shares add back.
- Converting to a smaller unit means multiplying: km m, kg g, ₹5.50 paise.
- Compare rates only in a common unit: ₹40 per kg beats ₹42 per kg, and km/h is m/s, so slower than m/s.
- Decide which direction "better" runs — lower for a price, higher for efficiency.
You will remember all of this far better after answering five questions on it than after reading it twice.
- To divide in the ratio , the shares are and — the denominator is the total parts.
- ₹1200 in gives ₹480 and ₹720, and in gives , and — always check the shares add back.
- Converting to a smaller unit means multiplying: km m, kg g, ₹5.50 paise.
- Compare rates only in a common unit: ₹40 per kg beats ₹42 per kg, and km/h is m/s, so slower than m/s.
- Decide which direction "better" runs — lower for a price, higher for efficiency.
You will remember all of this far better after answering five questions on it than after reading it twice.