Four ways to change a number

Every day you change numbers without noticing. You buy things, share things, count things in rows and work out how much is left. Almost all of it uses just four operations: adding, subtracting, multiplying and dividing.

This layer is about what each one means, not about long methods. By the end you should be able to read a little story, see the numbers in it, and say: "Ah, that is a sharing problem, so it is division", and then check that your answer makes sense.

Ammu's mother sends her to the kirana shop at the end of the lane with a ₹200 note. Here is what happens:

1. She buys a bag of rice for ₹64 and a packet of dal for ₹38. The shopkeeper, Ramesh uncle, works out the bill.
2. He gives Ammu her change from the ₹200 note.
3. Ammu also wants biscuits for her little brothers. Each packet costs ₹10 and she wants 4 packets, so she asks what that would cost.
4. At home, she has a jar of 24 toffees to share equally among 4 cousins.

Four little problems. Four different ways of changing numbers. Can you tell which is which before you read on?

Adding means putting amounts together to find how many there are altogether.

A city bus at the Charminar stop has 27 passengers. At the next stop 15 more climb on and nobody gets off. How many are on the bus now? We are joining a new group to the old one, so we add: 27 + 15 = 42.

The numbers being added are called addends, and the answer is the sum. In 27 + 15 = 42, the addends are 27 and 15, and the sum is 42.

Subtraction has two everyday meanings, and it helps to know both.

1. Taking away. You had 35 sweets and ate 12. How many are left? 35 − 12 = 23. Something is removed, and we find what remains.

2. Comparing (finding the difference). Riya is 138 cm tall and Arjun is 129 cm. How much taller is Riya? Nothing is taken away from anybody! We are asking how far apart two numbers are: 138 − 129 = 9 cm.

Either way, the answer to a subtraction is called the difference.

An egg tray at the market holds 30 eggs. How many eggs are in 4 trays?

You could add: 30 + 30 + 30 + 30 = 120. But because every tray holds the same number, there is a shortcut: multiply. 4 × 30 = 120.

That is the first big idea of multiplication: repeated addition of equal groups. We read 4 × 30 as "4 groups of 30" or "4 times 30". The answer is called the product.

The second picture is an array: things in neat rows and columns. For the school assembly, 6 rows of 8 chairs are set out.

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6 rows of 8 is 6 × 8 = 48 chairs. Now turn your head sideways: you see 8 columns of 6, which is 8 × 6. It is the same chairs, so it must be the same number, 48. An array shows at a glance why the order of multiplying does not matter.

The third picture is scaling, or "times as many". Meera has 12 stamps. Her brother has 3 times as many. That means three lots of Meera's collection: 3 × 12 = 36 stamps.

Words like double (× 2), triple (× 3) and "five times as much" are all multiplication.

Division also has two everyday meanings.

1. Sharing (fair shares). 18 chocolates are shared equally among 3 children. How many does each child get? Hand them out one at a time, round and round, until they are gone: each child gets 6. We know the number of groups and want the size of each share.

2. Grouping (how many groups?). A teacher has 36 pencils and puts 6 in each box. How many boxes can she fill? Keep taking out 6 at a time: she fills 6 boxes. We know the size of each group and want how many groups.

Both are written the same way, 36 ÷ 6 = 6. The number being divided (36) is the dividend, the number we divide by (6) is the divisor, and the answer (6) is the quotient.

Sometimes things do not share out perfectly. Nani has made 23 laddoos for 5 grandchildren. Each child gets 4 laddoos, which uses 4 × 5 = 20, and 3 are left over.

The amount left over is called the remainder. We write 23 ÷ 5 = 4 remainder 3, or 4 R 3.

A remainder is always smaller than the number you are dividing by. If 5 or more were left, you could give everyone one more!

Adding and subtracting are a pair: one undoes the other. If you add 6 to 9 you get 15; subtract 6 from 15 and you are back at 9. Multiplying and dividing are a pair in the same way.

Operations that undo each other are called inverse operations. Three numbers that belong together make a fact family:

Here is a question to ask of every word problem: what is happening in the story?

• Groups are being joined, or you want a total of different amounts → add.
• Something is taken away, or you are comparing two amounts, or finding how many more are needed → subtract.
• Equal groups are repeated, or something is "times as many" → multiply.
• Something is shared equally, or split into equal groups → divide.

Words such as "altogether", "left", "each" and "share" can be clues, but read carefully: the whole story matters more than any single word.

Before or after you calculate, ask: roughly how big should the answer be? A quick rough answer is called an estimate.

The easiest way to estimate is to round each number to the nearest ten (or hundred) first. To round to the nearest ten, look at the ones digit: 0–4, round down; 5–9, round up. So 48 → 50 and 31 → 30.

Estimate 48 + 31: 50 + 30 = 80. The exact answer is 79, very close. If you had written 709 or 17, the estimate would warn you straight away.

Notice how often multiplication turns up: whenever the same amount repeats, such as the same fare for every child or the same distance every hour, multiplying is the fast way to the total. Division shows up whenever people want things to be fair. Adding and subtracting keep track of totals and gaps.

Big real problems often need more than one operation. For a school trip, the teacher might multiply to find the total bus fare (₹450), add the cost of snacks, then subtract from the money collected to see what is left. You will meet many of these multi-step problems in the deeper layers.