Class 5 Mathematics Chapter 4: We the Travellers — II

A road trip begins

Nazrana and her friends are packing their bags. They will drive from Delhi to Mumbai. Then they go to Goa, then Hyderabad, and last of all Puri. Every hop has a number written on the map. How far will they travel in all? To find out, they must add big numbers. Let us learn how.

Fuel for the lorry

A lorry beside them is filling its tank. The tank already holds 28 litres of fuel. The driver fills 75 litres more. How much fuel is in the tank now? We add 28 and 75.

Ones under ones

Write one number under the other. Keep ones under ones. Keep tens under tens. Now add the ones. 8 and 5 make 13. That is 1 ten and 3 ones. Write 3 in the ones place. Carry the 1 ten across.

Trade ten for one

Now add the tens. 2 and 7 and the carried 1 make 10 tens. Ten tens make one hundred. So write 0 in the tens place. Write 1 in the hundreds place. The tank holds 103 litres. This trading is called regrouping.

The long road south

In 2019 the North-South Corridor joined Srinagar and Kanniyakumari. One town on it was 1,855 km from Srinagar. The same town was 1,862 km from Kanniyakumari. Add the two parts. 1,855 plus 1,862 gives 3,717. So the corridor was 3,717 km long.

Four digits, same steps

Big numbers do not need new rules. Start at the ones column. Move to the left, one column at a time. Carry whenever a column crosses nine. The steps never change, however long the number is.

Money adds the same way

Mahesh drives his family all the way south. He spends 21,880 rupees on fuel and toll tax. He spends 38,900 rupees on other things. Add the two amounts. The total is 60,780 rupees.

Drop the place labels

You need not write TTh, Th, H, T and O on top. Just keep the digits in a neat column. Try 267 plus 54. The 4 sits under the 7, so nothing slips. The answer is 321.

Adding the whole trip

Delhi to Mumbai is 1,600 km. Mumbai to Goa is 590 km. Goa to Hyderabad is 670 km. Hyderabad to Puri is 1,055 km. Add them one hop at a time. The friends drove 3,915 km.

Two friends: plus and minus

If 46 plus 21 makes 67, then 67 minus 21 gives 46. Also 67 minus 46 gives 21. One addition fact hides two subtraction facts inside it. This works for big numbers too.

Check every sum

Use that friend fact to check your work. Take your answer and subtract one of the numbers. You should get the other number back. A quick check catches a slipped digit.

Remember

• Line the digits up before you add.
• Always start with the ones column.
• Ten of one place make one of the next place.
• Check your sum by subtracting.

Scenario:

A bus travels 1,083 kilometres in one week and 1,367 the next. A class adds them. Ravi writes the numbers down and gets 2,350. Meena gets 24,100 because she lined the digits up wrongly, putting the thousands under the hundreds. Their teacher asks the class to check which answer is sensible. Two journeys of about a thousand each cannot reach twenty-four thousand. Meena finds her slip and lines the columns up again. She gets 2,450. The class compares the two careful answers and finds Ravi carried one ten short. The right total is 2,450.

Analysis:

One wrong answer came from crooked columns and one from a missed carry. Asking whether an answer is sensible catches the first kind immediately.

Activity:

Add two four-digit distances. Check your answer is sensible before you check the columns.

Title:

The family trip adding game

Safety:

Only paper and pencil are needed, so this is safe to do alone.

What You Notice:

Adding two at a time is easier than adding four at once, and the check tells you at once if a digit slipped.

A motorbike tank holds about 5 to 15 litres of fuel, but a train can take about 5,000 litres.

The longest road journey on Earth runs about 15,150 km, from Talon in Russia to Sagres in Portugal.

Where to start adding — Start on the right, then march left. — Always add the ones first. Then tens, then hundreds. Carrying only works in that order.

A ship leaves Mumbai

A cargo ship sails out of Mumbai port. It is bound for Chennai port, far to the south. The whole sea route is 2,700 km. On the way it stops at Cochin port. By then it has sailed 1,083 km. How much of the journey is still left? This is a taking away question.

The fuel meter jumps

First, a smaller problem. A minibus has 18 litres of fuel left in its tank. After filling, the meter shows 65 litres. How much fuel went in? We take 18 away from 65. The answer is 47 litres.

Borrow from the left

In 65 minus 18 the ones column will not work. You cannot take 8 from 5. So borrow one ten from the tens column. That ten becomes 10 ones. Now the ones column has 15. And 15 minus 8 leaves 7.

Finish the columns

After borrowing, the tens column holds 5 instead of 6. Take away the 1 ten of 18. That leaves 4 tens. So the answer is 47 litres. Borrowing is regrouping again, only the other way round.

How much sea is left

Now back to the ship. Take 1,083 away from 2,700. The ones and tens are zeros, so borrow across them. One hundred becomes ten tens. One ten becomes ten ones. The answer is 1,617 km still to sail.

Zeros need patience

A zero cannot lend anything by itself. Go left until you find a digit that can lend. Then pass the value back, one place at a time. Work slowly, column by column, and zeros stop being scary.

Two very long roads

The longest road route on Earth is 15,150 km. Africa's longest highway measures 10,228 km. Which is longer, and by how much? Take the smaller from the bigger. The gap is 4,922 km.

Drop the place labels

You need not write the place letters on top of a subtraction either. Keep the digits in a neat column. Try 654 minus 195. Borrow twice as you go. The answer is 459.

Always add back

Checking a subtraction is easy. Add your answer to the number you took away. You should get the first number back. For the ship, 1,617 plus 1,083 gives 2,700. That matches.

Mary counts her money

Mary starts a train journey with 12,540 rupees. She spends 3,275 rupees, so 9,265 rupees are left. An uncle gives her 4,900 rupees, making 14,165 rupees. Tickets take 2,645 rupees and gifts take 1,275 rupees. She ends with 10,245 rupees.

Enough, or short?

A mela printed 18,225 certificates for Chittoor and 18,104 children came. Take away and you get 121 spare. Jaunpur printed 19,043 but 19,265 children came. There the district fell 222 short.

Remember

• Subtract from the ones column upwards in place value.
• Borrow one from the left when a column is too small.
• A zero borrows from further left and passes it back.
• Check by adding your answer to the number taken away.

Scenario:

Farida works in the goods office of Mumbai port. Every ship that leaves has a route written in her ledger. On Monday a ship sets out for Chennai port, 2,700 km away by sea. It stops at Cochin port after 1,083 km to load spices. Farida writes down how far is still to go, so the office knows when the ship will arrive. The next ship carries 8,250 kg of rice, but only 6,525 kg is loaded. Farida notes the space still free. Ports in India move food, cloth and machines between states every single day. Her small subtractions keep the whole line of trucks and ships moving on time.

Analysis:

Subtraction answers the question how much is left. Ports, trains and shops all use it to plan loads, space and arrival times.

Activity:

Ask an adult the distance from your town to a city you like. Find out how far a bus travels in one hour, then work out how much of the journey would be left after that hour.

A port is where ships load and unload, just as a bus stand is where buses pick up people.

Priyanka Mohite has stood on five Himalayan summits above 8,000 metres, before any other Indian woman.

When to borrow — Too small on top? Knock on the left. — If the top digit is smaller than the bottom one, the column to its left lends you one unit worth ten.

Checking a subtraction — Add it back and see. — Your answer plus the number you took away must give the number you started with.

Two teams of numbers

Here are two teams. One team holds 1, 2, 7 and 9, and its total is 19. The other holds 3, 4, 5 and 9, and its total is 21. Can you make both totals the same by swapping one number from each team? Try it before you read on.

One swap can do it

Swap the 2 with the 3. The first team becomes 1, 3, 7 and 9, which makes 20. The second team becomes 2, 4, 5 and 9, which also makes 20. The teams were 2 apart, so a swap of 1 fixed it.

Numbers in a row

Numbers that follow each other with no gaps are called consecutive numbers. 1, 2, 3, 4 are consecutive. So are 29, 30, 31 and 2024, 2025, 2026. Runs like these hide neat patterns inside their sums.

Three in a row

Add 1, 2 and 3 to get 6. Add 2, 3 and 4 to get 9. Add 3, 4 and 5 to get 12. Look closely. Each answer is three times the middle number. So 67 plus 68 plus 69 must be three times 68, which is 204.

The step between sums

Now look at the sums 6, 9 and 12. Each one is 3 more than the last. With four numbers in a row, the sums are 10, 14 and 18, going up by 4. The step is simply how many numbers you added.

Jumping to a hundred

Sukanta wants to fill 32 plus a number to make 100. Going to 99 is easy, because no column needs regrouping. 32 plus 67 makes 99. One more step reaches 100, so the missing number is 68.

Filling a thousand

The same trick fills 1,000. To go from 877 to 999 you add 122. One more gives 1,000, so the missing number is 123. Try 666 in the same way. The missing number there is 334.

Taking away nine

Namita loves the number 9. To take 9 away, first take 10 away. You have taken 1 too many, so give that 1 back. For 67 minus 9, take 10 to reach 57, then add 1 to get 58.

Taking away ninety-nine

The same idea works for 99. Take away 100, then give 1 back. So 187 minus 99 becomes 187 minus 100, which is 87, and then 88. You can do this one in your head.

Numbers that read both ways

Nitin likes numbers that read the same forwards and backwards. They are called palindrome numbers. 22, 363 and 8,558 are palindromes. Between 100 and 200 there are exactly ten of them, starting at 101 and ending at 191.

Even and odd pairs

Line counters up in twos. An even number makes full pairs. An odd number leaves one counter alone. Two evens added give an even. Two odds added also give an even, because the two lonely counters pair up. An odd and an even give an odd.

Remember

• One swap can move a total up or down.
• Three numbers in a row add to three times the middle one.
• Jump to 99 or 999 first, then take one more step.
• To take away 9, take 10 and give 1 back.

Scenario:

A shopkeeper adds 29 and 31 in her head instantly. A class asks how. She says she moved one across. Taking 1 from 31 and giving it to 29 makes 30 and 30, which is 60. The children test it on 48 and 52. Moving 2 across makes 50 and 50, which is 100. Ravi checks both the long way and the totals match. He sees why the trick works. Moving an amount from one number to the other takes it off one and adds it to the other, so the total never changes.

Analysis:

The trick is not a short cut past the arithmetic. One number goes down by exactly what the other goes up, so the total is untouched.

Activity:

Add three pairs like 29 and 31 by moving an amount across. Check each one the long way.

Title:

The pairing tray

Safety:

Do not use small caps near a baby or a young child.

What You Notice:

Two odd groups always join into an even one, because the two lonely caps find each other and make a pair.

The year 2002 was a palindrome year. The next one will be 2112.

The sum of any two numbers in a row is always odd, because one of them is odd.

Taking away 9 or 99 — Take a ten, give a one back. — 9 is one less than 10 and 99 is one less than 100, so you always give one back.

Three numbers in a row — Middle times three. — The number below and the number above the middle balance each other out.