On this page
"If 6 workers build a wall in 10 days, how long will 15 workers take?" Many students multiply and get 25 days, which makes no sense: more workers should finish sooner. The skill in this chapter is not the arithmetic. It is deciding which kind of relationship you are looking at. This post shows how to tell direct from inverse proportion, works four examples, and ends with a short routine.
Where is proportion in the new Class 8 NCERT books?
The older NCERT book had one chapter, Direct and Inverse Proportions (Chapter 11, per its contents page). The new Ganita Prakash books split it in two:
Part I, Chapter 7, Proportional Reasoning-1: ratios, simplest form, problem solving with proportional reasoning, sharing in a ratio and unit conversions.
Part II, Chapter 3, Proportional Reasoning-2: a recap, ratios in maps, ratios with more than two terms, pie charts and inverse proportions.
How can you tell direct from inverse proportion?
The Part II chapter summary gives both definitions:
Direct proportion: both quantities change by the same factor, and their quotient stays the same. If x and y are directly proportional, x₁/y₁ = x₂/y₂ = k.
Inverse proportion: when one quantity changes by a factor n, the other changes by 1/n. Their product stays the same: x₁y₁ = x₂y₂ = k.
The chapter's own example is a trip from Lucknow to Kanpur. Walking at 5 km/h takes 18 hours, a bicycle at 15 km/h takes 6 hours, and a car at 60 km/h takes 1.5 hours. Speed × time is 90 km every time, which is the distance.
So the test is simple. Make a small table and ask: does the ratio stay constant, does the product stay constant, or neither?
Worked examples
Example 1 (direct). 12 notebooks cost ₹420. What do 20 cost?
Cost per notebook: 420 ÷ 12 = ₹35
20 notebooks: 20 × 35 = ₹700
As a proportion: 12 : 420 :: 20 : x, so x = 420 × 20 ÷ 12 = 700
Example 2 (inverse). 6 workers build a wall in 10 days. Working at the same rate, how long do 15 workers take?
More workers means fewer days, so this is inverse
The constant product is 6 × 10 = 60 worker-days
Days for 15 workers: 60 ÷ 15 = 4 days
The hidden assumption is that every worker works at the same rate and all can work at once. Real problems sometimes break that, so say it when you write the answer.
Example 3 (inverse). A 240 km trip at 60 km/h takes 4 hours. How long at 80 km/h?
Product: 60 × 4 = 240, the distance
Time at 80 km/h: 240 ÷ 80 = 3 hours
Check: 80 × 3 = 240
Example 4 (neither). Meera is 12 and her brother is 4. When Meera is 24, how old will he be?
The proportional answer, 8, is wrong
Both ages go up by the same amount, 12 years, so he will be 4 + 12 = 16
The difference (8 years) stays constant, not the ratio or the product
Why do students use proportion where it does not fit?
Because it is the method they have practised most. Van Dooren and colleagues (2005) gave proportional and non-proportional word problems to 1,062 students from Grades 2 to 8. Students often used proportional methods where they clearly did not apply. These errors increased up to Grade 5, as students got better at proportion itself. From Grade 6 students told the cases apart more often, but even in Grade 8 a considerable number of proportional errors remained.
The fix is to ask what stays constant before you calculate, every time.
Which mistakes come up most often?
Using direct when it is inverse. Workers and days, speed and time, and pipes filling a tank are usually inverse.
Turning one ratio upside down. In 12 : 420 :: 20 : x, keep notebooks on the left and rupees on the right in both ratios.
Ignoring assumptions. "Same rate" and "same distance" are what make the product constant.
Mixing units. 1.5 hours is 90 minutes, not 1 hour 50 minutes. Part I, Chapter 7 has a section on unit conversions for this.
Forcing proportion on additive change. Ages, as in Example 4, grow by addition.
A five-day practice routine
Day 1: Write ten everyday situations and label each direct, inverse or neither before any calculation.
Day 2: For each, build a three-row table and check whether the quotient or the product stays constant.
Day 3: Solve a shuffled set of eight word problems: three direct, three inverse, two neither.
Day 4: Do five unit-conversion problems, then redo any wrong ones from Day 3.
Day 5: Draw a pie chart, where each angle is directly proportional to its count. Our post on pie charts and data handling in Class 8 shows how.
Percentages are the next step in proportional thinking; see our guide to comparing quantities in Class 8. If you keep choosing the wrong method, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning rather than printing an answer, so you can see why a problem is direct or inverse.
Before your next homework problem, write one line first: "Constant: ratio, product, or neither."
