Another Peek Beyond the Point

Why This Matters

You go to a shop. One pen costs ₹9.50. You buy 5 pens. How much do you pay?

You buy 2.250 kg of oranges. The price is ₹56.50 for one kilogram. What is your bill?

A ribbon is 3.9 metres long. You cut it into 10 equal pieces. How long is each piece?

A car goes 126 km in 2.5 hours. How fast was it going?

Every one of these is a real question you will face in daily life. And every one of them needs you to multiply or divide decimals — numbers with a point in them.

In the last chapter you learned what decimals are. You learned to read them, compare them, and add and subtract them. Now we take the next step. We learn to multiply and divide them.

Here is the good news. You already know how to multiply and divide whole numbers. Multiplying and dividing decimals is the same skill, with one extra job: getting the decimal point in the right spot. And the best part — we will see exactly why the point goes where it goes. Nothing here is a “just do it” rule. Every rule has a reason, and you will see the reason with a picture.

The Big Idea

A decimal is just a fraction in disguise. 0.3 means 3/10. 0.07 means 7/100. So multiplying or dividing decimals is really multiplying or dividing fractions — and we already know how to do that. The whole trick is this: do the sum with the digits as if there were no point, then put the point back in the right place. Counting the right place is never a guess. The place value chart tells you exactly where it goes, every single time.

Let’s Break It Down

Before anything else, let’s refresh what a decimal really stands for. Everything in this chapter rests on it, so let’s be sure it’s solid.

Multiplying and dividing a decimal by 10, 100, 1000 — the point shifts, and WHY

Let’s start with the easiest and most useful skill: multiplying or dividing by 10, 100, or 1000.

Take 2.5 × 10. Let’s not guess. Let’s use the fraction trick.

2.5 = 25/10. So 2.5 × 10 = (25/10) × 10 = 250/10 = 25.

Look at what happened. 2.5 became 25. The digits 2 and 5 stayed the same, in the same order. But the answer is 10 times bigger.

Now 2.5 ÷ 10. Again: 2.5 = 25/10. Dividing by 10 means 25/10 ÷ 10 = 25/100 = 0.25.

Same digits 2 and 5, in the same order, but now 10 times smaller.

So when you multiply or divide by 10, the digits don’t change — they just move to a different place on the place value chart. It only looks like the point is moving. Figure 4.1 shows exactly what is going on.

Tap to zoom

Why does it work this way? Each column on the chart is 10 times bigger than the column on its right. A digit in the ones column is worth 10 times the same digit in the tenths column. So if you make the whole number 10 times bigger, every digit jumps up one column (to the left). And if you make it 10 times smaller, every digit drops one column (to the right).

Now the simple shortcut. We usually keep the digits still and move the point the other way instead. The result is the same number:

Multiply by 10 → move the point 1 place to the right.

Multiply by 100 → move the point 2 places to the right.

Multiply by 1000 → move the point 3 places to the right.

The number of zeros in 10, 100, 1000 tells you how many places to move. For dividing, you move the point the other way — to the left:

Divide by 10 → move the point 1 place to the left.

Divide by 100 → move the point 2 places to the left.

Divide by 1000 → move the point 3 places to the left.

A few quick examples so the pattern sinks in:

• 5.7 × 10 = 57 (point moves 1 right)
• 5.7 × 100 = 570 (point moves 2 right; we add a zero to make room)
• 0.306 × 1000 = 306 (point moves 3 right)
• 3.9 ÷ 10 = 0.39 (point moves 1 left)
• 3.9 ÷ 100 = 0.039 (point moves 2 left; we add a zero in front)
• 24 ÷ 100 = 0.24 (point moves 2 left)

Notice we sometimes add zeros — at the end when multiplying (570) or at the front when dividing (0.039) — just to have a digit for each place. Those extra zeros are place-holders.

Let’s make sure the direction is clear, because mixing it up is the most common slip here.

Multiplying a decimal by a whole number

Now let’s multiply a decimal by an ordinary whole number, like 9.5 × 5.

The slow-but-sure way: multiplying by 5 just means adding 9.5 to itself 5 times.

9.5 + 9.5 + 9.5 + 9.5 + 9.5 = 47.5.

This is exactly the pen problem from the start. One pen is ₹9.50, and 5 pens cost ₹47.50. Figure 4.2 shows the five jumps on a number line.

Tap to zoom

Adding five times works, but it is slow. Here is the fast way, using the fraction trick.

9.5 = 95/10. So 9.5 × 5 = (95/10) × 5 = 475/10 = 47.5.

See the shortcut hiding inside? We multiplied 95 × 5 = 475 (just the digits, no point), then placed the point so the answer had one digit after it — because 9.5 had one digit after the point. That gives 47.5.

So the rule is: drop the point, multiply the whole numbers, then put the point back so the answer has the same number of decimal places as the decimal you started with.

Here is the same idea on a slightly bigger example, the school-walk problem from your textbook.

Ajay walks 0.827 km to school and 0.827 km back, so that is 0.827 × 2 each day:

0.827 = 827/1000, so 0.827 × 2 = (827 × 2)/1000 = 1654/1000 = 1.654 km a day.

Over 6 days a week: 1.654 × 6 = (1654 × 6)/1000 = 9924/1000 = 9.924 km a week.

Multiply the digits (1654 × 6 = 9924), then keep 3 decimal places (because 1.654 had 3). Simple.

Multiplying a decimal by a decimal (count the decimal places — and WHY)

Now the big one: multiplying two decimals, like 0.3 × 0.2.

A lot of students guess the answer is 0.6. It is not. The real answer is 0.06. Let’s see why, because once you see it, you will never get it wrong again.

Use the fraction trick on both numbers:

0.3 = 3/10 and 0.2 = 2/10.

To multiply two fractions, multiply the tops and multiply the bottoms:

(3/10) × (2/10) = (3 × 2)/(10 × 10) = 6/100 = 0.06.

Look at the bottom. 10 × 10 = 100. We started with two “over 10” fractions, so the answer is “over 100” — which means two digits after the point. That is the whole secret. Figure 4.3 shows it as a picture you can actually see.

Tap to zoom

Do you see it? 0.3 is a slice of the width. 0.2 is a slice of the height. Their product is the little rectangle where the two slices overlap. That rectangle is just 6 tiny squares, and each tiny square is one hundredth. So the answer is 6 hundredths = 0.06.

Now the general rule. When you multiply two decimals:

1. Ignore the points. Multiply the two numbers as whole numbers.
2. Count the decimal places in both original numbers and add them up.
3. Put the point in the answer so it has exactly that many decimal places.

Why does adding the places work? A decimal with 1 place is “over 10”. A decimal with 2 places is “over 100”. When you multiply two such fractions, the bottoms multiply: 10 × 100 = 1000. And 1000 has 3 zeros = 3 decimal places = 1 place + 2 places. The zeros always add up, so the decimal places always add up.

Figure 4.4 walks through a bigger product step by step, so you can copy the method.

Tap to zoom

Let’s do one fully, the way you would in your notebook.

Here is a thinking shortcut that uses the same idea. Suppose someone tells you 596 × 248 = 147808. Can you write down 5.96 × 24.8 straight away, without redoing the multiplication? Yes! The digits are identical (147808). You only need to place the point: 2 + 1 = 3 places, so the answer is 147.808.

Let’s check that the “add the places” idea is really clear.

One surprising thing about multiplying decimals: the answer is not always bigger than the numbers you started with.

With whole numbers, multiplying always makes things bigger (5 × 3 = 15, bigger than both). But 0.3 × 0.2 = 0.06 is smaller than both 0.3 and 0.2! Why? Because taking “0.2 of” something means taking only a fifth of it — and a part of a thing is smaller than the thing. The table below sums up when the product grows and when it shrinks.

The two numbers	Example	The product is...
Both bigger than 1	3.4 × 6.5 = 22.1	bigger than both numbers
Both between 0 and 1	0.75 × 0.4 = 0.3	smaller than both numbers
One bigger than 1, one between 0 and 1	0.75 × 5 = 3.75	between the two numbers

Dividing a decimal by a whole number

Now let’s divide. Start with dividing a decimal by an ordinary whole number, like the ribbon problem: 3.9 ÷ 10 we already did (= 0.39). But what about dividing by a number that is not 10, 100, or 1000 — like 9.5 ÷ 4?

A shopkeeper has 9.5 kg of sugar to pack equally into 4 bags. Each bag gets 9.5 ÷ 4 kg.

We use long division, the same method you know for whole numbers, with one new move: when you run out of whole units to share, you keep going into the tenths, then hundredths, and so on. The moment you cross the point, you write a point in your answer.

There is also a neat trick for some divisions: turn the divisor into 10, 100, or 1000 by finding an equivalent fraction. For example, 29 ÷ 4 = 29/4. Since 4 × 25 = 100, multiply top and bottom by 25:

29/4 = (29 × 25)/(4 × 25) = 725/100 = 7.25.

This works whenever the bottom is a factor of 10, 100, or 1000 (like 2, 4, 5, 8, 25, 50). When it is not (like dividing by 3), use long division instead.

Dividing by a decimal

Last skill: dividing by a decimal, like 126 ÷ 2.5 (the speed of the car).

Dividing by a number with a point is awkward. So we use a clever trick: turn the divisor into a whole number first.

The divisor 2.5 has one digit after the point. Multiply it by 10 and it becomes 25 — a whole number. But we can’t change just one number, or the answer would change. So we multiply the other number by 10 as well:

126 ÷ 2.5 = (126 × 10) ÷ (2.5 × 10) = 1260 ÷ 25 = 50.4.

So the car’s speed was 50.4 km per hour. Figure 4.5 shows the same trick on another example.

Tap to zoom

Why is it allowed to multiply both numbers? Because a division is really a fraction, and you can multiply the top and bottom of a fraction by the same number without changing its value. 6 ÷ 2 = 3, and 60 ÷ 20 = 3 too. The answer stays put. So 126 ÷ 2.5 and 1260 ÷ 25 are the same answer.

How many times do you multiply by 10? Count the decimal places in the divisor. If the divisor has 1 place (like 2.5), multiply both by 10. If it has 2 places (like 0.13), multiply both by 100. Look:

4.68 ÷ 0.13 = (4.68 × 100) ÷ (0.13 × 100) = 468 ÷ 13 = 36.

One more surprise to notice. When you divide by a decimal smaller than 1, the answer comes out bigger than the number you started with. For example, 128 ÷ 0.4 = 320, and 320 is bigger than 128! That feels strange, but it makes sense: 0.4 fits into 128 many, many times — more than 128 itself would.

Real-life money and length problems

You now have every tool. Let’s use them on the kinds of questions that actually come up. Watch how each one is just “multiply or divide decimals, then read the answer in the right units.”

The shopping problem from the start: Meenu buys 4 notebooks at ₹15.50 each and 3 erasers at ₹2.75 each. Figure 4.6 lays out the full bill.

Tap to zoom

See the plan: multiply for each item, then add the totals (lining up the points, just like Chapter 3). Total = 62.00 + 8.25 = ₹70.25.

A length-into-different-units example: a rupee coin is 1.45 mm thick. Stack 36 of them. How tall is the stack, in centimetres?

First the height in mm: 1.45 × 36. Drop the points: 145 × 36 = 5220. Then 2 decimal places: 52.20 mm.

Now change mm to cm. There are 10 mm in 1 cm, so we divide by 10: 52.20 ÷ 10 = 5.22 cm. The stack is 5.22 cm tall.

Common Mistakes

These are the slips that trip up almost everyone. Read each one — knowing the trap is half the battle.

Quick Check

Try these. Each one checks a “why,” not just an answer.

Practice Problems

Try each one yourself first. Then reveal the full solution.

Easy

Medium

Challenge

Summary

• A decimal is a fraction in disguise: a decimal with 1 digit after the point is “over 10”, with 2 digits “over 100”, with 3 digits “over 1000”. Counting the digits after the point tells you the bottom number.
• Multiplying by 10, 100, 1000 moves the point that many places to the right (the number gets bigger). Dividing moves the point the same number of places to the left (the number gets smaller). The digits never change — only the place they sit in.
• To multiply a decimal by a whole number, drop the point, multiply, then give the answer the same number of decimal places as the decimal you started with.
• To multiply two decimals, drop both points, multiply the digits, then add the decimal places of both numbers and place the point so the answer has that many places. (This is because the “over 10s” multiply: 10 × 100 = 1000.)
• Multiplying decimals does not always give a bigger answer. If both numbers are between 0 and 1, the product is smaller than both.
• To divide a decimal by a whole number, use long division; when you run out of whole units, regroup into tenths, then hundredths — and place the point in the answer the moment you cross it.
• To divide by a decimal, multiply both numbers by 10, 100, or 1000 to make the divisor a whole number, then divide normally. This keeps the answer the same because you scaled the top and bottom equally.
• In real life, the pattern is always: multiply or divide the decimals, then read the answer in the right units (rupees, kilograms, centimetres), converting units with ×10 or ÷10 when needed.

What’s Next

You can now do every basic operation with decimals — add, subtract, multiply, and divide. Next, in Chapter 13 — Connecting the Dots, you will use these number skills inside a different setting. Just as decimals let you measure quantities precisely, the ideas ahead will help you spot how separate pieces of information connect and link up — turning the careful arithmetic you have built here into reasoning about whole situations.