In-text — Recognising Inverse Proportion — Proportional Reasoning-2

1. 13 marksGanita Prakash Cl-8 Part 2, Math Talk, page 63

   Puneeth's father rides from Lucknow to Kanpur in 3 hours at 30 km/h. If he drives at 60 km/h instead, can the problem be written as the statement of proportionality 30 : 60 :: 3 : x? Will the travel time increase or decrease as the speed increases?

   Hint. Solve the proposed statement and check whether the answer makes sense in the real situation.

   Test the proposed statement by solving it. If 30 : 60 :: 3 : x were correct, the rule of three would give x = (60 × 3) ÷ 30 = 6 hours.

   That answer says doubling the speed doubles the travel time — that going faster takes longer. It is plainly wrong, so the statement of proportionality is wrong.

   Why it is wrong. Writing a : b :: c : d assumes a direct proportion, in which both quantities grow together. Speed and time for a fixed journey do not behave that way. The distance from Lucknow to Kanpur is fixed, so the faster you go, the less time you need. As speed increases, travel time decreases.

   What the correct statement looks like. For an inverse proportion the products match rather than the cross-products: speed₁ × time₁ = speed₂ × time₂ 30 × 3 = 60 × x 90 = 60x x = 1.5 hours

   Doubling the speed halves the time, which is what everyday experience says it should.

   The general form. If the relation had to be written as a proportion, the second ratio must be turned upside down: 30 : 60 :: x : 3, giving x = (30 × 3) ÷ 60 = 1.5 ✓ This flip is exactly what distinguishes inverse proportion from direct.

   ✦ No, 30 : 60 :: 3 : x is not correct — it gives 6 hours, meaning the faster trip takes longer. Speed and time are inversely proportional here, so travel time decreases as speed increases, and the correct working 30 × 3 = 60 × x gives 1.5 hours.

2. 23 marksGanita Prakash Cl-8 Part 2, Math Talk, page 64

   The table gives speed 5, 15, 30, 60 km/h with travel times 18, 6, 3, 1.5 hours. Does the time decrease by the same factor by which the speed increases? Check this for all the modes of transport, and say what stays constant.

   Hint. Compare each pair of columns: work out the factor for the speeds and the factor for the times.

   Compare the columns pair by pair.

   From → to	Speed factor	Time factor
   Walk → Bicycle	15 ÷ 5 = ×3	18 ÷ 6 = ÷3 ✓
   Bicycle → Motorcycle	30 ÷ 15 = ×2	6 ÷ 3 = ÷2 ✓
   Motorcycle → Car	60 ÷ 30 = ×2	3 ÷ 1.5 = ÷2 ✓
   Walk → Car	60 ÷ 5 = ×12	18 ÷ 1.5 = ÷12 ✓

   In every case the time is divided by exactly the factor the speed is multiplied by. So yes — the two quantities change by the same factor, but in opposite directions. That is what makes this an inverse proportion rather than a direct one.

   What stays constant. Multiply each pair: 5 × 18 = 90 15 × 6 = 90 30 × 3 = 90 60 × 1.5 = 90 The product is always 90, and it has a meaning: it is the distance from Lucknow to Kanpur in kilometres, which of course does not change with the mode of transport.

   The general statement. Two quantities x and y vary in inverse proportion if xy = k for some constant k. Equivalently, if x₁y₁ = x₂y₂, then dividing gives x₁/x₂ = y₂/y₁ — the ratio of the x-values equals the reversed ratio of the y-values. Checking that with walking and the car: 5/60 = 0.0833… and 1.5/18 = 0.0833… ✓

   ✦ Yes — the time is divided by exactly the factor the speed is multiplied by, so the quantities are inversely proportional. What stays constant is the product speed × time = 90, which is the fixed distance in kilometres.

3. 33 marksGanita Prakash Cl-8 Part 2, Math Talk, page 66

   Ram cuts a given quantity of vegetables in 1 hour and Shyam takes 1.5 hours for the same quantity. Working out how long they take together, we compare 5/3 units of work with 1 unit of work. Is the quantity of work and the time taken to complete it directly or inversely proportional? Find how long they take together.

   Hint. Ask what happens to the time if the amount of work is doubled while the working rate stays the same.

   Answer the proportionality question first. For a fixed working rate, doing twice as much work takes twice as long, and half as much work takes half as long. Work and time move up and down together, so they are directly proportional.

   This is worth separating from the inverse relationships elsewhere in this section. Workers and time for a fixed job are inversely proportional — more workers, less time. Work and time at a fixed rate are directly proportional — more work, more time. It is the quantity being held constant that decides which.

   Now the calculation. Treat the whole job as 1 unit of work and find what each does in one hour. Ram finishes in 1 hour, so in 1 hour he does 1 unit. Shyam finishes in 1.5 hours, so in 1 hour he does 1 ÷ 1.5 = 2/3 unit.

   Together in one hour they do 1 + 2/3 = 5/3 units

   Scale down to one unit. Since work and time are directly proportional, 5/3 units : 1 unit :: 1 hour : x hours (5/3) × x = 1 × 1 x = 1 ÷ (5/3) = 3/5 hour

   Convert for sense. 3/5 hour = (3/5) × 60 = 36 minutes.

   Check that the answer is reasonable. Two people working together must finish faster than the quicker of them alone, and 36 minutes is indeed less than Ram's 60 minutes. If the answer had come out longer than 1 hour, the rates would have been added the wrong way round.

   ✦ Work and time are directly proportional at a fixed rate. Together Ram and Shyam do 5/3 units per hour, so one unit takes 3/5 hour = 36 minutes.

Solutions written by the tuition.in editorial team and checked against the NCERT Class 8 Mathematics textbook Ganita Prakash Part 2, Reprint 2026-27 (hegp203.pdf), where this is Chapter 3 (pages 55-69) — the tenth chapter of the Class 8 course. Like the rest of Part 2, this PDF carries NO printed answer key, so every answer was derived from first principles and independently recomputed in Python before being written. TWO FIGURE-ONLY ITEMS WERE RESOLVED BY MEASURING THE RENDERED PAGE rather than guessing: (1) the eight sleep ring charts on page 67 were rendered at 300 dpi and the shaded arc of each ring measured by sampling 3600 points around its circumference — the eight measured fractions map one-to-one onto the eight values the question offers (giraffe 2.96 -> 2.5, elephant 3.84 -> 3.5, human 7.78 -> 8, dog 10.07 -> 10.5, cat 12.99 -> 13, squirrel 15.08 -> 15, python 18.09 -> 18, bat 20.16 -> 20), which is what fixes the assignment; (2) the transport pie chart on page 68 prints only four of its five angles, so the page was rendered and the labels read off before deducing Car = 30 degrees. ONE ITEM IS FLAGGED AS UNANSWERABLE FROM THE PDF: the map distances in section 3.2, because the chapter PDF stamps its map 'Map not to scale' - the method is given and no distance is invented. ONE APPARENT SLIP IN THE BOOK IS FLAGGED: page 68 question 5(iii) asks how many children use 'taxis', but taxi is not one of the five modes in the pie chart and the five slices already total 360 degrees, so the answer as printed is zero; the solution says so and notes that 'two-wheeler' (36 children) was probably intended.. Questions are referenced from the NCERT textbook for identification.