Add and subtract three-digit numbers

Line up equal places

A library has 248 books and receives 175 more. Find 248 + 175. Write hundreds under hundreds, tens under tens and ones under ones.

Start with ones. 8 + 5 = 13: exchange ten ones for one ten and keep three ones. Now there are 4 + 7 + 1 = 12 tens. Exchange ten tens for one hundred and keep two tens. Finally 2 + 1 + 1 = 4 hundreds.

The same column working records every exchange. Read the answer digits from hundreds to ones.

A 1 sent on means one unit of the next larger place. The next row includes it once. For 998 + 2, each column keeps a zero and sends one unit on. The last exchange becomes the new leading 1 in 1000; do not drop it.

The total is 423. The exchanged ten and hundred are part of the total, not extra amounts to add again.

Estimate first: 248 is near 250 and 175 is near 180, so a total near 430 is sensible. A result of 323 would lose a hundred.

Subtract through a zero

Find 503-178. Five hundreds, no tens and three ones cannot give away eight ones yet.

Exchange one hundred for ten tens: four hundreds, ten tens, three ones. Exchange one of those tens for ten ones: four hundreds, nine tens, thirteen ones. The amount is still 503.

In these written columns, “available” is the amount after exchanges and before subtracting the digit in that place. The tens have already passed one unit to the ones, leaving nine tens.

The last row takes nothing from above the hundreds. Every exchange preserves the original amount; it does not add another hundred or ten.

Now subtract each place: 13-8 = 5 ones, 9-7 = 2 tens and 4-1 = 3 hundreds. The difference is 325.

Check by adding: 325 + 178 = 503. If the check fails, look for an exchange you used twice or forgot.

Check your understanding 1 Keep the amount unchanged

Find 402-156. First show how many hundreds, tens and ones are available after exchanging. Complete the written subtraction on paper and use addition to check your answer.

Hundreds after exchanging

Tens after the exchanges

Ones after exchanging

Difference

Check answer

Show answer

Regroup 402 as three hundreds, nine tens and twelve ones. Then 12-6 = 6, 9-5 = 4 and 3-1 = 2, giving 246. Check 246 + 156 = 402. The number check compares the regrouping and result; use this model to check your full written subtraction too.

Choose a short mental route

For 364 + 20, keep the hundreds and ones: 364 + 20 = 384. For 364-100, one hundred is removed: 264. For 364 + 5, add the ones to get 369.

A jump can cross a place boundary. For 398 + 5, use two to reach 400, then add the three left: 403. For 403-5, take away three to reach 400, then two more: 398.

Check your understanding 2 Find an unknown starting amount

A jar has some beads. You add 126 beads and now have 384. How many were there at first? Explain why you chose your operation.

Your answer or drawing on paper

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The starting amount plus 126 equals 384. The unknown is a part, so subtract: 384-126 = 258. Check 258 + 126 = 384. Adding 126 again would find a different amount, not the starting one.

The written method and a mental method should agree. Choose the one that lets you keep track of every part accurately.