Maths shelf · Learning guide · Class 5

The Magic of 11 at Dadi's Chai Stall

Eleven cups of chai at 13 rupees each; the trick writes 1, then 1 plus 3, then 3, giving 143

Eleven cups of chai at ₹13 each. Tanmay says "₹143" without even picking up a pencil. How?

about 15 minutes of reading and puzzles, then about 35 minutes for the activity
Classes 4–7
Learning guide

What you will learn: a quick way to multiply by 11, why it works, when it breaks and how to fix it, and how to check any answer with a quick estimate.

Small test at the end: scroll down to the end of this page for five short questions that check what you have learnt. Tap an answer, then press Check answer. Go to the questions ↓

Jump to: The story · Now try it · Check yourself · For grown-ups · For teachers

Safety first

Nothing special: paper, a pencil and a ruler.

Keep it safe:

Use a calculator only for the final check, if you have one.

Eleven cups, in a flash

It was Sunday, and the gully cricket team had just beaten the team from the next lane. All 11 players crowded round Dadi's chai stall.

"Eleven cutting chais, please!" shouted the captain. "How much?"

One cup costs ₹13. Tara reached for a pencil to work out 13 × 11.

Tanmay didn't even look up. "₹143," he said.

The captain counted out the coins. Tara stared at her brother. "How did you do that so fast? Did you cheat?"

"It's the magic of 11," said Tanmay, grinning.

Dadi put down her kettle. "There's no magic in maths, only reasons. Let's find this one."

1. The trick

Tanmay showed them on the back of a bill.

To multiply a two-digit number by 11:

Write the first digit.
In the middle, write the two digits added together.
Write the last digit.

13 × 11 → 1, (1 + 3), 3 → 143
25 × 11 → 2, (2 + 5), 5 → 275
42 × 11 → 4, (4 + 2), 2 → 462

Tara checked 42 × 11 the long way. 462. It worked!

The trick: for a two-digit number × 11, put the sum of its digits in the middle.

2. Why it works

"But why does it work?" asked Tara. "Tricks you don't understand are like borrowed slippers. They fall off when you run."

Dadi smiled. "Think of 11 as 10 + 1. So 25 × 11 means 25 tens plus 25 ones."

25 × 10 = 250
25 × 1 = 25
250 + 25 = 275

"Now look at the addition in columns," said Dadi.

25 × 11 is 250 + 25; in the column addition, the middle column adds the two digits 5 and 2 to make 7, giving 275

"The middle column holds the 5 from 250 and the 2 from 25. That's why the middle is the two digits added (as long as they add up to less than 10; more on that soon). It isn't magic. It's just place value."

The rule: × 11 is the same as × 10, then add the number once more. The trick is a shortcut for that addition.

3. When the trick breaks

The next Sunday, the team ordered samosas at ₹58 each.

This time Tara was ready, because now she knew why the trick worked. "5, then 5 + 8 = 13, then 8... ₹5138!"

The captain nearly fainted. "Five thousand rupees? For samosas?"

Tanmay shook his head. "Tara, use your common sense. ₹58 is about ₹60, and 60 × 11 is 660. The answer should be near 660, not 5000!"

Tara went red. "So the trick broke?"

"The trick didn't break," said Dadi. "It just needs a carry. Remember the column addition? When a column adds up to 10 or more, you carry 1 into the next column."

58 × 11: the middle 5 + 8 is 13, so write 3 and carry 1 to the 5, giving 638, which matches 580 + 58

5 + 8 = 13, which is too big for one place.
Write the 3 in the middle and carry the 1 to the front: 5 + 1 = 6.
58 × 11 = 638.
Check: 580 + 58 = 638. ✔ And 638 is near our estimate of 660. ✔

The rule: if the middle sum is 10 or more, write its last digit and carry 1 to the front digit.

4. Your turn: drive a number you've never seen

Use the rule and check each answer with an estimate.

34 × 11 = ?
76 × 11 = ?
99 × 11 = ?
Dhruv wrote 47 × 11 = 4117. Without working it all out, how can you tell it's wrong? What is the right answer?
Challenge: Dadi says the same idea works for 123 × 11. Use 11 = 10 + 1 to find out.

Check your answers

374: 3, (3 + 4 = 7), 4. Check: 340 + 34 = 374.
836: 7 + 6 = 13, so write 3 and carry 1: 7 + 1 = 8. That gives 8, 3, 6. Check: 760 + 76 = 836.
1089: 9 + 9 = 18, so write 8 and carry 1: 9 + 1 = 10. That gives 10, 8, 9 = 1089. Check: 990 + 99 = 1089.
47 is about 50, and 50 × 11 = 550, so 4117 is far too big. The right answer is 517: 4 + 7 = 11, so write 1 and carry 1: 4 + 1 = 5. That gives 5, 1, 7. Check: 470 + 47 = 517.
1353: 1230 + 123 = 1353. The pattern is 1, (1 + 2), (2 + 3), 3: each neighbouring pair of digits is added.

Fun fact: the 1s make a staircase

11 × 11 = 121
111 × 111 = 12321
1111 × 1111 = 1234321

The digits climb up and come back down. Can you work out why, using columns? (Hint: the pattern stops working once a column reaches 10, at 1111111111 × 1111111111.)

Where the story stops being true

There is no magic in maths, even when it feels like magic. Every trick is a shortcut for a reason, and here the reason is 11 = 10 + 1.
The "middle sum" trick only fits 11. For × 12 you would need a different shortcut (12 = 10 + 2).
A trick is one good way, not the only way. Long multiplication is just as correct, only slower.

🧠 Logic corner

You just used two thinking tools:

Deduction (rule → new case): from the rule "11 = 10 + 1" you worked out answers for numbers the story never showed, even three-digit ones.
The sense check (estimating): Tanmay caught ₹5138 because he knew the answer had to be near 660. Checking whether an answer is reasonable catches mistakes that no trick can.

Start here (Class 4)

Begin with numbers whose digits add up to less than 10: 12, 23, 34, 45. Check every answer the long way, with 10 + 1. Leave the carry until these feel easy.

Go further (Classes 6–7)

Use the same idea, 11 = 10 + 1, on three-digit numbers: 123 × 11, 352 × 11, 789 × 11. Each neighbouring pair of digits is added, with a carry wherever a pair adds up to 10 or more. Then try × 9 as 10 − 1, and × 99 as 100 − 1. Which shortcut is the easiest to explain?

Words to know

Digit: one of the signs 0 to 9 that we use to write numbers. 58 has two digits, 5 and 8.
Place value: what a digit is worth because of where it sits. In 58, the 5 means 5 tens.
Carry: when a column adds up to 10 or more, the ten moves into the next column as 1.
Estimate: a quick, rough answer, made by rounding, to see what size the real answer should be.
Sense check: asking "Is this answer a sensible size?"

🧪 Now try it

The Chai Stall Price Board

about 35 minutes

You need: paper, a pencil, a ruler; a calculator only for the final check

You make: a "Team of 11" price board, worked out three ways

A price board table: samosa ₹15, estimate about 150, 165 both ways; lassi ₹38, estimate about 440, 418 both ways; kulfi ₹46, estimate about 550, 506 both ways

The safety notes for this activity are at the top of the page.

Notice
Make a price board for a pretend stall with six items, each priced between ₹10 and ₹99 (for example, samosa ₹15, lassi ₹38, kulfi ₹46, sandwich ₹67, juice ₹74, thali ₹89). Which prices have digits that add up to 10 or more?
Grown-up: "Before you work anything out, which items do you think will be the trickiest for the ×11 trick?"
Teacher aim: Reading two-digit numbers by place value and spotting when a carry will be needed.
Guess
A cricket team of 11 walks in, and everyone orders the same item. For each item, write a quick estimate first (round the price, then × 11).
Grown-up: "What's a sensible size for the answer? Could it be in the thousands?"
Teacher aim: Estimating before calculating, so the answer can be sense-checked.
Try
Work out each total two ways: with the ×11 trick (carry when the middle is 10 or more), and with × 10 + the price once more.
Grown-up: Let the child work without the calculator; save it for the end.
Teacher aim: Using a shortcut alongside the method it comes from (the distributive idea: 11 = 10 + 1).
Observe
Make a table: item, estimate, trick answer, ×10 + once answer, calculator check. Circle any row where two answers disagree.
Grown-up: "Where did the two methods disagree? Which step went wrong?"
Teacher aim: Recording results and finding errors from the evidence, not by guessing.
Explain
Pick the trickiest item and explain to someone why the carry is needed, using columns.
Grown-up: "Teach it to me as if I've never seen the trick."
Teacher aim: Explaining a procedure through place value, not only following it.
Ask another question
Try × 101: 23 × 101, 47 × 101, 85 × 101. What pattern do you see? Can you explain it with 100 + 1?
Grown-up: "Could you invent a trick for × 9 using 10 − 1?"
Teacher aim: Generalising: 23 × 101 = 2300 + 23 = 2323. Every shortcut like this comes from splitting a number into easy parts.

Done when: your board shows every item worked out three ways, with any disagreements explained.

Make it on paper. Keep it with you. Nothing to upload.

Same spark, other subjects

Dadi's chai stall

Prices, sums and the words people use when they buy and sell.

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For grown-upsThe safety notes are at the top of the page.

Play "chai stall" at home: one person sets a price, and the other works out 11 cups using the trick, then checks with an estimate.
When a child gets a wrong answer, ask "Is that a sensible size?" before "What did you do?".
Praise the check as much as the answer.

For teachersLearning aims, links and what to look for.

Connects to: place value, multiplying by 10, the distributive idea (11 = 10 + 1), carrying in column addition, and estimation. These come up in the multiplication and number work of Classes 4–6, and the "number tricks" style of NCERT's Class 6 Ganita Prakash. A teacher should match it to the exact chapter their class is using.
Look for three things:
- The child uses the trick correctly, including the carry.
- The child can explain why it works with 10 + 1 and columns.
- The child checks answers with an estimate.
Class variation: groups invent a "trick" for × 9 (9 = 10 − 1) and test whether it always works.

These links are general. They have not been checked against a specific board syllabus.