Compensation when adding and subtracting

When you reach for it

Aulani runs the school store with her class. The store has 457 pencils on the shelf, and a box of 299 more arrives. How many pencils does the store have now?

The answer is 457 + 299. You could line the numbers up and add by place value, trading ones for a ten and tens for a hundred as in regrouping in addition. But 299 is very close to 300, and adding 300 is easy: only the hundreds digit changes. So add 300 instead, then fix the answer:

Adding 300 added one pencil too many, so one is taken back. The store has 756 pencils.

This move is called compensation: change a number into a round one that is easy to work with, then make up for the change. You reach for it when one of the numbers sits just below or just above a multiple of ten or a hundred, such as 299, 198, 401 or 99, and you want the answer in your head. Prices that end in 99 cents are made for it.

The lesson has three parts, one for each day. Part 1 is compensation in addition. Part 2 is compensation in subtraction, along with a second way to keep a difference the same. Part 3 uses both on money.

The procedure

Part 1: adding a round number, then taking the extra back

The picture. Think of the sum on an open number line: a line with only the numbers you need marked on it, not drawn to scale (as in adding on an open number line). Start at 457. Adding 299 means moving 299 to the right. Make that move in two hops: a big hop of 300 to land on 757, and then a small hop of 1 back to 756.

Together the two hops move 300 - 1 = 299 to the right, which is exactly the move that was asked for. So the landing point, 756, is the sum.

Why the extra is taken back. The number 299 is 1 less than 300. When you add 300, you have added the 299 you wanted plus 1 more. That extra 1 has to come off:

The same thinking works with any amount. To add 198, add 200 and take 2 back, since 200 is 2 more than 198. To add 395, add 400 and take 5 back. The amount you take back is always the gap between the number you were given and the round number you used.

Rounding down instead. A number can also sit just above a round number. To add 301, add 300 and then add 1 more, because 300 is 1 short of 301. So here the fix goes the other way: if the round number was too big, take the difference off; if it was too small, add the difference on.

Give and take: a second way. Instead of fixing the answer at the end, you can fix the other number first. Move 1 from the 457 over to the 299, so that 299 becomes 300:

Nothing was added and nothing was lost; one pencil only changed piles. The step is allowed by the properties of addition (properties of addition). Write 457 as 456 + 1. Then

Moving the parentheses is the associative property of addition: when you add three numbers, you may group them either way and the sum is the same. The 1 that came out of 457 joined the 299 and made it round.

Which number to round. Either addend may be the one made round, since the order of addends does not matter (the commutative property of addition). Pick the one closer to a round number. In 457 + 299, the 299 is 1 away from 300, while 457 is 43 away from 500, so round the 299.

Part 2: subtracting a round number, then giving the extra back

The picture. Morven needs 503 - 198. By place value, this needs a trade across the zero in 503 (see regrouping in subtraction and subtracting three-digit numbers by place value). Compensation avoids the trade. Taking away 200 is easy, so on the number line start at 503 and hop back 200 to land on 303.

But only 198 was to be taken away, and 200 is 2 more than 198. The hop went 2 too far to the left, so hop forward 2 to land on 305:

Why the fix goes the other way here. In addition, rounding up adds too much, so you take some off. In subtraction, rounding up the number being taken away takes away too much, so you put some back. Think of 503 pencils on a shelf. Taking away 200 pencils removes 2 more than asked, and putting those 2 back on the shelf leaves the right number. The rule underneath is the same as in Part 1: undo the change you made. The change was “2 too many taken away”, and the way to undo it is to add 2.

If the number being taken away sits just above a round number, the fix is the other way again. For 815 - 302, take away 300 to get 515. That took away 2 too few, so take away 2 more: 513.

Same difference: a second way. A difference is the distance between two numbers on the number line: 503 - 198 is how far it is from 198 up to 503. If both numbers slide up by the same amount, the distance between them does not change. Slide both up by 2, so that 198 becomes the round number 200:

In the figure the two bars are drawn to scale and have the same length. The bottom one has only moved 2 to the right. This way needs no fixing at the end, because the change was made to both numbers at once.

Notice how this differs from give and take in addition. For a sum, one number goes up and the other goes down by the same amount, so the total stays. For a difference, both numbers go up (or both go down) by the same amount, so the distance stays.

Part 3: money in the head

How a price is written. A dollar is 100 cents (see Money). A price tag such as 2.99 dollars is read “2 dollars and 99 cents”: the number before the point counts whole dollars, and the two digits after it count cents. Here the point is only a way of writing dollars and cents side by side; every sum below is also done in cents.

A price of 99 cents is 1 cent less than a dollar. So a price such as 99 cents is the round amount 1 dollar, less 1 cent, and 2.99 dollars is 3 dollars less 1 cent. Stores choose these prices often, and each one is an invitation to compensate.

Seonaid buys a folder for 99 cents and a glue stick for 65 cents at the school store. Think of the folder as 1 dollar. Then the two cost 1 dollar and 65 cents, and taking back the 1 cent leaves 1 dollar and 64 cents. In cents only, it is the same kind of sum as Part 1:

That is 164 cents. Since 100 cents make 1 dollar, 164 cents is 1 dollar and 64 cents, written 1.64 dollars, which is what she pays.

Spending. Aulani has 4.20 dollars and spends 99 cents on the folder. Take away a whole dollar to leave 3 dollars and 20 cents. That took away 1 cent too many, so give it back: 3 dollars and 21 cents, or 3.21 dollars. In cents, 420 - 99 = 420 - 100 + 1 = 321.

Every move from Parts 1 and 2 works with money, because an amount of money is a number of cents. Working in whole dollars plays the part that working in whole hundreds played before.

Worked examples

Boundaries

• Undo the change in the right direction. In addition, a round number bigger than the addend adds too much, so the gap is taken off; a round number smaller than the addend adds too little, so the gap is added on. In subtraction, a round number bigger than the number being taken away takes away too much, so the gap is added back. The test is always: what did the round number do that the real number would not have, and what undoes it?
• Undo the gap, not the round number. The fix is the distance between the given number and the round one: for 457 + 299 it is 1, the gap between 299 and 300.
• Opposite ways for a sum, the same way for a difference. Give and take keeps a sum only when one addend goes up and the other goes down by the same amount. Same difference keeps a difference only when both numbers move the same way by the same amount. Moving the two numbers of a difference in opposite ways changes it: 503 - 198 = 305, but 501 - 200 = 301.
• It helps only near a round number. When neither number is close to a round hundred, as in 436 + 254, rounding to a hundred leaves a big gap to fix, and the move saves nothing. Partial sums, place value, or compensating to a nearby ten instead are then the quicker way.
• Whole numbers all the way. Within whole numbers, the first step of take-away compensation must not take away more than you have. For 199 - 197, taking away 200 from 199 is not possible with whole numbers. Same difference still works: add 3 to both, and 199 - 197 = 202 - 200 = 2.
• Money: 100 cents to a dollar. Compensating with money treats each whole dollar as a round number, and the fix is counted in cents. When cents reach 100 they make a dollar, as in 164 cents = 1.64 dollars.

Where this goes next

Compensation is the same idea as doubling and halving: turn the problem into an easier one and keep track of what the change did. Here it lets you add and subtract within 1{,}000 without writing, and it gives a second way to check an answer found by partial sums or by place value. The same habit of reaching for a round number is the start of estimating sums and differences, where the round numbers are kept and the fix is skipped, and of rounding to the nearest ten or hundred. Later it works just as well with bigger numbers and with decimals: 4.8 + 2.7 is 5 + 2.7 - 0.2.

Exercises

Part 1: compensation in addition

1. Find 399 + 245 by adding 400 and taking the extra back.
2. Find 198 + 527.
3. Find 463 + 297.
4. Find 588 + 99.
5. Find 156 + 498.
6. Find 301 + 482. Round the 301 down to 300 first.
7. Fill in the boxes using give and take: 395 + 238 = 400 + \square = \square.
8. The school store has 279 pencils, and a box of 199 more arrives. How many pencils does it have now?

Part 2: compensation in subtraction

9. Find 734 - 299 by taking away 300 and giving the extra back.
10. Find 861 - 198.
11. Find 520 - 396.
12. Find 700 - 497.
13. Find 958 - 601. Take away 600 first.
14. Fill in the boxes by sliding both numbers up: 433 - 197 = \square - 200 = \square.
15. The book fair has 815 books, and 398 are sold. How many are left?

Part 3: money

16. Add 99 cents and 47 cents. Give the total in dollars and cents.
17. A ruler costs 2.99 dollars and a pack of markers costs 3.40 dollars. What do they cost together?
18. At the book fair, Aulani buys two books, one for 4.99 dollars and one for 2.99 dollars. What is the total?
19. Seonaid has 6.50 dollars and spends 1.99 dollars at the school store. How much does she have left?
20. Aulani has 9.15 dollars and buys a book for 4.98 dollars. How much does she have left?
21. Morven buys three erasers at the school store, each for 99 cents. Find 99 + 99 + 99 cents in your head, and give the cost in dollars and cents.

Reasoning

22. Morven finds 562 + 199 by adding 200 first. Should he then add 1 or take 1 away? Explain, and find the sum.
23. Explain, using a number line, why 812 - 295 and 817 - 300 have the same answer. Then find it.
24. Neither number in 436 + 254 is close to a round hundred. Would rounding one of them to a hundred help much here? Explain, and find the sum another way.
25. Explain why the price 1.99 dollars plus the price 2.99 dollars costs 2 cents less than 5 dollars, and give the total.