Proportional Reasoning – 2

Why This Matters

You make these decisions every single day, often without noticing.

You are cooking for two friends, but suddenly four more turn up. How much more rice and dal do you need? You are riding a bicycle to school, but today you take a bus instead. Will you reach faster, and by how much? Three people are painting a wall and it is taking too long, so two more join in. How many days will it take now?

Each of these is a question about how one quantity changes when another changes. Sometimes the two go up together — more friends, more food. Sometimes one goes up while the other goes down — more painters, fewer days.

This chapter teaches you to spot the difference, and to put exact numbers on it. Once you can, you will never again guess “how much” — you will calculate it. That single skill quietly powers cooking, shopping, travel, building, and even reading a map.

The Big Idea

Two quantities can be linked in two opposite ways. In direct proportion, they grow together by the same factor — double one and the other doubles, so their quotient (y ÷ x) stays fixed. In inverse proportion, one grows as the other shrinks by the same factor — double one and the other halves, so their product (x × y) stays fixed. The whole chapter is learning to tell which case you are in, and then solving it with simple multiply-and-divide steps.

Let’s Break It Down

Before we start, let us quickly refresh the idea of a proportional relationship from Proportional Reasoning – 1, because everything here builds on it.

Checking if two ratios are proportional: cross-multiplication

Reducing both ratios to simplest form works, but there is a faster test. It is called cross-multiplication.

Suppose Viswanath makes idli batter with 6 cups of rice and 3 cups of dal (ratio 6 : 3). Puneet uses 4 cups of rice and 2 cups of dal (ratio 4 : 2). Will their idlis taste the same? They will, if the two ratios are proportional.

Here is the trick. Write the ratios as two fractions: 6/3 and 4/2. Multiply across the equals sign — top-left times bottom-right, and bottom-left times top-right. If both answers match, the ratios are proportional.

The picture below shows the two cross products meeting in the middle.

Tap to zoom

So the general rule is short and worth remembering.

Two ratios a : b and c : d are proportional when their cross products match:

a × d = b × c

Notice why this works. Saying a/b = c/d is the same as saying the two fractions are equal. To compare two fractions, you bring them to the same value by cross-multiplying — that is just the fraction rule you already know, applied to ratios.

Direct proportion: growing together

Now to the heart of the chapter. We start with the case where two quantities grow together.

Think again about idli batter. The recipe needs rice and dal in the ratio 2 : 1. If you want more batter, you add more of both, keeping that ratio. The table below shows a few amounts.

For 2 cups of rice you use 1 cup of dal. For 4 cups of rice, 2 cups of dal. For 6, then 3. For 8, then 4. Look what stays the same: in every row, dal ÷ rice = 1/2.

This is direct proportion. When one quantity goes up, the other goes up by the same factor. Double the rice, the dal doubles too. Triple it, the dal triples too.

Here is the key fact, and the “why” behind it. Because both quantities always change by the same factor, their ratio never changes. So if you divide one by the other, you always get the same number. We call that fixed number the constant of proportionality, written k.

Two quantities x and y are in direct proportion when their quotient is always the same:

y ÷ x = k (a constant)

Equivalently, y = k × x.

There is a beautiful way to see direct proportion. If you plot the pairs (rice, dal) as points on a graph, they all land on a straight line that passes through the origin (the corner point 0, 0). Figure 3.2 shows both the table and this graph together.

Tap to zoom

Why does the line pass through the origin? Because if you use 0 rice, you use 0 dal. No batter at all. That point (0, 0) is always on a direct-proportion graph, and the steady ratio keeps the line straight.

The everyday way to use direct proportion is the unitary method — “unit” meaning one. You first find the value for one unit (by dividing), then scale up to however many you want (by multiplying).

Let us see it on a real problem.

Figure 3.3 lays out those two steps as a simple flow, so the “find one, then find many” habit sticks.

Tap to zoom

Inverse proportion: trading off

Now the opposite case — and the one students most often confuse. Sometimes when one quantity goes up, the other goes down.

Here is the classic example. Puneeth’s father travels from Lucknow to Kanpur. The distance is fixed. He can go at different speeds, and the time changes. The table below shows it.

Walking at 5 km/h takes 18 hours. By bicycle at 15 km/h it takes 6 hours. By motorcycle at 30 km/h, 3 hours. By car at 60 km/h, just 1.5 hours.

Look carefully. As the speed goes up, the time goes down. Faster means you spend less time on the road. That makes good sense — if you move quicker, the same journey is over sooner.

But there is something more precise hiding here. Going by bicycle (15 km/h) is 3 times faster than walking (5 km/h), because 15 ÷ 5 = 3. And the time drops from 18 hours to 6 hours — exactly 3 times smaller, because 18 ÷ 6 = 3. The speed went up by a factor of 3, and the time went down by the same factor of 3.

This is inverse proportion. When one quantity changes by a factor n, the other changes by the inverse factor 1/n (the upside-down version). Up by 3 on one side, down to 1/3 on the other.

And here is the “why” that makes it click, plus the key fact. In every row of the table, multiply speed by time:

5 × 18 = 90

15 × 6 = 90

30 × 3 = 90

60 × 1.5 = 90

Every product is 90. That number is the distance between the two cities (90 km) — and the distance never changes, no matter how you travel. That is why the product stays fixed.

Two quantities x and y are in inverse proportion when their product is always the same:

x × y = k (a constant)

So in inverse proportion the product is fixed (unlike direct, where the quotient is fixed). If you know two pairs of values (x₁, y₁) and (x₂, y₂), then because both products equal the same k:

x₁ × y₁ = x₂ × y₂

This one equation solves almost every inverse-proportion problem. You will use it again and again.

There is also a tidy way to see inverse proportion on a graph. If you plot the pairs, they do not make a straight line. They make a downward curve that swoops from high values down towards the axes but never touches them. Figure 3.4 shows the table and this curve side by side, so you can compare it with the straight line of direct proportion.

Tap to zoom

Let us solve an inverse problem with the product rule.

The same product rule cracks pumps filling a tank, food provisions lasting a number of days, and machines making toys. Whenever “more of one means less of the other,” reach for x₁ × y₁ = x₂ × y₂.

Telling the two apart

So the whole game is: is this direct or inverse? Get that right and the rest is easy. Figure 3.5 puts the two side by side so the contrast is sharp.

Tap to zoom

A quick test you can run in your head: ask “when the first quantity goes up, does the second go up or down?”

• Goes up too → direct (the quotient y ÷ x is fixed).
• Goes down → inverse (the product x × y is fixed).

Here is the side-by-side summary in a table.

A trickier one: people working together

Sometimes a problem mixes the ideas. Suppose Ram can cut a pile of vegetables in 1 hour, and Shyam can cut the same pile in 1.5 hours. If they work together, how long will it take?

The clever move is to think in terms of work done in one hour, not total time. Count the whole job as 1 unit of work.

• Ram finishes in 1 hour, so in 1 hour he does 1 unit of work.
• Shyam finishes in 1.5 hours, so in 1 hour he does 1 ÷ 1.5 = 2/3 units of work.

Together, in one hour they do 1 + 2/3 = 5/3 units. Now this last step is a direct proportion: the amount of work done is directly proportional to the time spent. More time, more work — both go up together.

Common Mistakes

These three slips cost the most marks. Read them once and dodge them for good.

Quick Check

Try these four. Each one tests an idea from above.

Practice Problems

Try each one yourself first. Only then tap to see the full solution.

Easy

Medium

Challenge

Summary

• A ratio compares quantities (like rice : dal = 2 : 1); two ratios are proportional when they describe the same relationship.
• To test if two ratios are proportional, use cross-multiplication: a : b and c : d are proportional when a × d = b × c.
• In direct proportion, both quantities change by the same factor (double one, the other doubles). The quotient y ÷ x = k stays constant, and the graph is a straight line through the origin.
• In inverse proportion, one quantity goes up as the other goes down by the same factor. The product x × y = k stays constant (so x₁ × y₁ = x₂ × y₂), and the graph is a downward curve.
• To tell them apart, ask: when the first quantity goes up, does the second go up (direct) or down (inverse)?
• The unitary method solves direct problems: find the value for one unit by dividing, then multiply up.
• For “working together” problems, add up the work each does in one hour, then use direct proportion to find the total time.

What’s Next

You can now read the relationship between two changing quantities and put exact numbers on it — a skill you will use far beyond this classroom, every time you cook, shop, travel, or build.

Next we return to shapes. The following chapter is Exploring Some Geometric Themes, where you will look more closely at angles, lines, and the neat patterns that geometry hides in plain sight. See you there!