Class 5 Mathematics Chapter 6: The Dairy Farm

Daljeet's butter trays

Daljeet Kaur runs a small milk unit. Today her butter packets sit in trays. One tray has 3 rows of 2 packets. Her helper turns the tray around. Now it looks like 2 rows of 3 packets. Did the packets change? Let us count and find out.

Turn the tray

Count the first way. That is 3 × 2 = 6 packets. Count the second way. That is 2 × 3 = 6 packets. The tray is the same tray. Only our way of looking changed. Turning a tray never adds or removes packets.

Any order works

Daljeet has 6 groups of 13 packets. So she has 6 × 13 = 78. Her helper counts 13 groups of 6. He gets 13 × 6 = 78. Both answers match. You may multiply two numbers in any order. Pick the order that feels easier.

Zero packets

One tray is empty. It has 9 rows of 0 packets. So 9 × 0 = 0. Turn it around and 0 × 9 = 0 too. Zero groups hold nothing at all. Any number times zero gives zero.

The ten trick

A crate holds 10 packets. Four crates hold 4 × 10 = 40 packets. Twenty crates hold 20 × 10 = 200 packets. Look at the two answers. Each one is the first number with a zero added at the end.

Why a zero appears

Multiplying by 10 makes a number ten times bigger. Every digit moves one place to the left. The ones place is then empty. We write 0 there to hold the place. That is why 4 grows into 40.

Hundreds and thousands

Multiplying by 100 adds two zeros. So 3 × 100 = 300 and 8 × 100 = 800. Multiplying by 1,000 adds three zeros. So 5 × 1,000 = 5,000. In the same way, 20 × 1,000 = 20,000. The pattern keeps going.

Two tens meet

What is 30 × 20? Split both numbers. It is 3 × 10 × 2 × 10. Now group them in a new order. You get 3 × 2 = 6 and 10 × 10 = 100. So the answer is 6 × 100 = 600.

Count the zeros

Here is a quick rule. Multiply the front numbers first. Then write back every zero you set aside. For 60 × 50, do 6 × 5 = 30. You had set aside two zeros. So 60 × 50 = 3,000.

The hidden zero

Now try 40 × 50. First do 4 × 5 = 20. Then write back the two zeros you kept. You get 2,000, not 200. The zero inside 20 counts as well. Always write every zero you have.

Daljeet checks her stock

Daljeet has 220 trays. Each tray holds 20 packets. She does 22 × 2 = 44 in her head. Two zeros were set aside. So she has 4,400 packets. A big sum turned into an easy one.

Remember

• Two numbers may be multiplied in any order.
• Any number times zero is zero.
• 10 adds one zero, 100 adds two, 1,000 adds three.
• Multiply the front numbers, then write back every zero.

Scenario:

A dairy stacks milk crates in rows. A class counts 6 rows of 4 crates and writes 24. Then the crates are restacked as 4 rows of 6. Meena expects a different total and counts again. It is still 24. Their teacher asks the class to try two more pairs. 3 rows of 8 and 8 rows of 3 both give 24. Meena sees that the order of the two numbers does not change the product. The class then adds an empty shelf, 5 rows of 0, and finds the answer is 0 however it is turned round.

Analysis:

Restacking the same crates cannot change how many there are. That is why the two numbers can be multiplied in either order.

Activity:

Arrange 24 small objects into three different row patterns. Write the multiplication for each one.

Title:

Build your own butter tray

Safety:

Caps and seeds are small. Keep them away from babies and toddlers.

What You Notice:

The two answers always match. Turning the rows never changes how many caps there are.

India makes more milk than any other country in the world.

A number multiplied by 1 never changes. So 1 × 745 is 745.

Multiplying numbers that end in zeros — Front first, zeros last. — Multiply the numbers in front of the zeros. Then write back every zero you set aside.

Five children, one sum

The teacher writes 18 × 5 on the board. Five children answer, and every child works in a different way. Each one still gets 90. How can so many paths end at the same place? Let us look at their ideas one by one.

Nida halves it

Nida says half of 18 is 9. She works out 9 × 5 = 45. Then she works out the other half, which is 45 again. She adds 45 and 45 to get 90. Two smaller sums felt easier than one big one.

Kanti splits it

Kanti breaks 18 into 10 and 8. She knows 10 × 5 = 50. She also knows 8 × 5 = 40. Adding them gives 50 + 40 = 90. Splitting a number into friendly parts is a strong habit.

John goes past and comes back

John thinks 18 is close to 20. He does 20 × 5 = 100. But he has counted two extra fives. So he takes away 2 × 5 = 10. That leaves 100 − 10 = 90.

Mili halves and doubles

Mili halves the 18 and doubles the 5. That turns 18 × 5 into 9 × 10. Ten times anything is easy, so she writes 90 at once. The numbers changed, but the answer did not.

Why does that work?

Take 3 × 18. Double the 3 to get 6. Halve the 18 to get 9. Now 6 × 9 = 54, and 3 × 18 = 54 too. One side of the sum grows twice as big. The other side shrinks to half. The two changes cancel out.

Pick the right one to halve

For 22 × 5, halve the 22 and double the 5. You get 11 × 10 = 110. Always halve the even number. Halving 5 would give a half, which is harder to use.

This trick loves 5 and 25

Try 16 × 25. Halve and double once to get 8 × 50. Do it again to get 4 × 100. That is 400. Doubling 25 makes 50, then 100, and those are lovely numbers.

The nearest multiple

Now try 4 × 19. Nineteen is nearly 20. So do 4 × 20 = 80. You have counted one extra four. Take it away: 80 − 4 = 76.

Sometimes you add back

Try 14 × 21. Twenty-one is one more than 20. So do 14 × 20 = 280. You are short by one fourteen. Add it back: 280 + 14 = 294.

Friendly hundreds

Try 99 × 15. Use 100 × 15 = 1,500. You counted one extra fifteen. So take away 15 and get 1,485. In the same way, 199 × 63 is 12,600 − 63 = 12,537.

Remember

• Split a number into friendly parts and add the results.
• Halve one number and double the other.
• Halve the even number, never the odd one.
• Go to a round number, then add back or take away.

Scenario:

A milk seller works out the cost of 18 litres at 5 rupees a litre. He splits 18 into 10 and 8. Ten litres cost 50 and eight cost 40, so the total is 90. A class checks it the long way and gets 90 too. Ravi then splits 18 into 9 and 9, giving 45 and 45, which is also 90. Their teacher asks which way is right. Both are. The class sees that one multiplication can be reached by several routes. Splitting is useful when the parts land on times tables you already know.

Analysis:

Three routes and one answer. Splitting is chosen for convenience, not correctness, so any split that lands on known tables will do.

Activity:

Work out 16 times 5 three different ways by splitting 16. Check all three answers match.

Title:

The two-ways race

What You Notice:

The clever ways finish first, and the answers always match. 5 × 18 is 90, 25 × 12 is 300, 15 × 22 is 330, 99 × 8 is 792 and 12 × 21 is 252.

Shopkeepers use the nearest-multiple trick every day for prices like ₹99 and ₹199.

Doubling 25 gives 50, and doubling again gives 100. That is why 25 is so easy to work with.

Halving and doubling — Halve the even one, double the other. — Halving an odd number makes a half, which is hard to use. So pick the even number to halve.

Nearest multiple — Jump to the round number, then step back. — Use 20 instead of 19 or 100 instead of 99. Then take away the extra groups you counted.

Milk from 268 villages

A dairy cooperative buys milk from 268 villagers. Each villager keeps at least 4 milk-giving cows. How many cows send milk to this dairy? The sum is 268 × 4. That looks big. Let us break it into pieces we already know.

Break the big number

Every number is made of place-value parts. So 268 is 200 + 60 + 8. Now multiply each part by 4. That gives 800, then 240, then 32. Three small sums have taken the place of one big one.

Put the parts back

Add the three answers: 800 + 240 + 32 = 1,072. So the cooperative gets milk from at least 1,072 cows. Splitting by place value always works. The parts are easy, and adding them is easy too.

Nida draws a box

Nida likes to draw a box grid. For 12 × 35, she writes 30 and 5 along the top. She writes 10 and 2 down the side. Now she has four small boxes to fill in.

Fill the four boxes

The top row gives 10 × 30 = 300 and 10 × 5 = 50. The bottom row gives 2 × 30 = 60 and 2 × 5 = 10. Every box holds one easy product. No box is hard.

Add the boxes up

Now add all four numbers: 300 + 50 + 60 + 10. That comes to 420. So 12 × 35 = 420. The box shows where each part of the answer comes from.

John uses fewer rows

John does not draw boxes. He writes 2 × 35 = 70 on one line. Then he writes 10 × 35 = 350 on the next. He adds the two lines to get 420. Same answer, fewer steps.

A bigger multiplier

The dairy has 453 Gir cows. Each one gives 13 litres of milk a day. John's way splits 13 into 10 and 3. First, 3 × 453 = 1,359. Then 10 × 453 = 4,530. Adding gives 5,889 litres.

Mind the tens row

The row for 10 × 453 must end in a zero. It is 4,530, never 453. Ten times a number is ten times bigger. Losing that zero is the most common slip of all.

Check, check!

Pankaj works out 203 × 54 and writes 135. That cannot be right. 200 × 50 alone is 10,000. His parts lost their place values. The true answer is 10,962.

Estimate first, then trust

A quick estimate guards your work. For 453 × 13, think 450 × 13 = 5,850. Your real answer, 5,889, sits right beside it. If the two are far apart, look again.

Remember

• Split each number into its place-value parts.
• Multiply every part, then add the results.
• The tens row always ends in a zero.
• Estimate first, so a silly answer cannot hide.

Scenario:

The Sharma family lives in Nagpur. There are four of them. Every month their kitchen throws out about 35 kg of waste. Peels, leaves and tea grounds all go into one bin. Instead of sending it to the dump, they make compost in a clay pot on the terrace. Each year the pot gives them about 150 kg of rich compost. They spread it on their tulsi, curry leaf and tomato plants. In the market, one kilogram of compost sells for ₹24. Their neighbours have started asking how the pot works. The children keep a small notebook of the weights, month by month.

Analysis:

Multiplying place-value parts turns a kitchen habit into a number, and shows how waste at home becomes soil and savings.

Activity:

Weigh your kitchen waste for one week. Multiply by 52 to estimate a whole year.

Gir cows come from Gujarat and are known for giving plenty of milk.

Village dairy cooperatives collect milk from thousands of small farmers every morning.

The tens row — Tens row, zero below. — When you multiply by the tens digit, the answer ends in a zero. So 10 × 453 is 4,530.

Order of work — Split, multiply, add. — Break the numbers into place-value parts, multiply every part, then add all the parts together.