Friendly numbers and deliberate over and under estimates

When you reach for it

Nayeli, Satoshi and Wenjing run their school’s food drive. Cans arrive by the hundred, get packed into boxes and go out on trucks. Wenjing also places the school’s order of supplies: boxes, tape, labels and markers. All day they need answers like “about how many cans go on each truck?” and “is 60 dollars enough?” An exact answer takes paper and time. An estimate takes a few seconds, as long as the numbers are easy to work with.

You already know one way to estimate: round each number to a place, then compute (Estimating by rounding). That works well for adding. For dividing and multiplying it often leaves numbers that are still awkward. This lesson adds three ideas, one for each part.

• Compatible numbers, also called friendly numbers: numbers close to the real ones, chosen so that the computation becomes a basic fact you know.
• Benchmarks: easy landmark numbers, such as 0, \tfrac{1}{2} and 1, or 25, 50 and 100, that other numbers are compared with.
• Overestimates and underestimates: knowing whether an estimate is above or below the exact answer, how far off it is, and when that matters.

The procedure

Part 1: numbers chosen to fit a fact

Seven trucks will share 4{,}186 cans equally. About how many cans go on each truck?

Try plain rounding first. To the nearest thousand, 4{,}186 becomes 4{,}000. But 4{,}000 \div 7 is not a fact you know, because 7 does not go evenly into 40. To the nearest hundred, 4{,}186 becomes 4{,}200, and that one does work. The rounding was lucky.

Compatible numbers turn the luck into a choice. Think of 4{,}186 as about 41 hundreds. Which numbers of hundreds does 7 divide evenly? The multiples of 7: 35, 42, 49 hundreds, and so on. The one nearest to 41 hundreds is 42 hundreds, which is 4{,}200.

So

because 42 \div 7 = 6, and 42 hundreds divided by 7 is 6 hundreds. Each truck takes about 600 cans.

Why is this allowed? A quotient changes only a little when the dividend (the number being divided) changes a little. Moving 4{,}186 to 4{,}200 adds 14 cans in all, which is only 2 more on each of the 7 trucks. The estimate stays close, and the arithmetic becomes a fact you already know.

For multiplication, compatible numbers are factors whose product is easy. The best-known pair is 25 and 4, because 25 \times 4 = 100. Others are 50 and 2, 20 and 5, and 125 and 8, which make 100, 100 and 1{,}000. Tens and hundreds are compatible with everything, since multiplying by them only adds zeros.

There is no single right choice. Two people may pick different compatible numbers and both be right. The test is the same for everyone: the new numbers stay close to the real ones, and the computation becomes easy.

Part 2: benchmarks

A benchmark is a number used as a landmark. For whole numbers the landmarks are the tens, hundreds and thousands, and handy numbers such as 25 and 50. For fractions between 0 and 1 the landmarks are 0, \tfrac{1}{2} and 1.

To tell which landmark a fraction is near, compare its numerator (the top number) with its denominator (the bottom number).

• The numerator is small next to the denominator, as in \tfrac{1}{9}. The fraction is near 0.
• The numerator is about half the denominator, as in \tfrac{5}{12}, where half of 12 is 6. The fraction is near \tfrac{1}{2}.
• The numerator is close to the denominator, as in \tfrac{7}{8}. The fraction is near 1.

The number line shows why the test works. \tfrac{5}{12} is one twelfth short of \tfrac{6}{12}, which is \tfrac{1}{2}. \tfrac{7}{8} is one eighth short of \tfrac{8}{8}, which is 1. Each is close to its landmark, so a sum of such fractions is close to the sum of the landmarks.

Decimals use the same landmarks. 0.48 and 0.52 are close to 0.5, which is \tfrac{1}{2}. 2.97 is close to 3, and 6.51 is close to 6.5. Finding a decimal’s nearest whole number or half is rounding a decimal to a convenient value. Halves are worth keeping because they pair up: 2.5 + 0.5 = 3, and \tfrac{1}{2} of a number is just half of it.

A number bigger than 1, such as 2\tfrac{7}{8}, is handled in two pieces: the whole number 2 stays, and the fraction \tfrac{7}{8} goes to its benchmark 1, so 2\tfrac{7}{8} is near 3.

Part 3: which way, and how far

An estimate is an overestimate when it is larger than the exact answer, and an underestimate when it is smaller. Often you can tell which, without knowing the exact answer, by watching which way each number moved.

• Adding. Round every number up, and the total can only grow, so the estimate is an overestimate. Round every number down, and it is an underestimate.
• Multiplying whole numbers. If both factors move up, the product grows. If both move down, it shrinks. If one moves up and the other down, the moves alone do not say which way the estimate is off.
• Dividing. A bigger dividend gives a bigger quotient: more cans means more per truck. A bigger divisor gives a smaller quotient: the same cans shared among more trucks means fewer per truck.

So 4{,}186 \div 7 \approx 600 is an overestimate. The dividend moved up and the divisor stayed the same.

The error of an estimate is how far it is from the exact answer: the larger of the two minus the smaller. Sometimes the error can be found without doing the exact computation. Moving 4{,}186 up to 4{,}200 added 14 cans, and 14 cans shared among 7 trucks is 2 per truck. So 600 is exactly 2 too many, and

Check by multiplying: 7 \times 598 = 4{,}186. This works because sharing 4{,}200 - 14 cans is the same as sharing the 4{,}200 and then taking back each truck’s part of the 14.

Whether an error matters depends on the question. Off by 2 cans out of about 600 is nothing when planning trucks. It is not fine on a label that promises an exact count.

The direction matters most when the question is “is there enough?” To be sure the money covers a bill, overestimate the cost: if even the bigger number fits the budget, the real cost does too. To be sure everyone has a seat, underestimate the seats or overestimate the people. An estimate that leans the wrong way cannot answer an “enough” question, however close it is.

Worked examples

Examples 1 to 3 belong to Part 1, compatible numbers. Examples 4 and 5 belong to Part 2, benchmarks. Examples 6 and 7 belong to Part 3, over and under estimates.

Boundaries

• Compatible numbers must stay close. A basic fact is only useful if the numbers it uses are near the real ones. Replacing 1{,}615 with 2{,}000 makes the division easy and the estimate poor, as Example 2 shows.
• An estimate is not the answer. Write \approx, never =, between a computation and its estimate. When an exact count is needed, compute it; the estimate then checks that the exact answer has the right size.
• The direction rules are for sums, products and quotients of positive numbers. In a difference, raising the number being subtracted makes the result smaller. Rounding both numbers up does not give an overestimate of a difference: 58 - 21 \approx 60 - 30 = 30, but the exact difference is 37.
• Mixed moves give no direction. When one factor moves up and the other down, the estimate may land on either side. 31 \times 58 \approx 30 \times 60 = 1{,}800 is just above the exact 1{,}798, while 39 \times 21 \approx 40 \times 20 = 800 is below the exact 819.
• A fraction exactly between two benchmarks, such as \tfrac{1}{4} between 0 and \tfrac{1}{2}, is equally close to both. Either may be used, as long as you know which way the estimate leans.

Where this goes next

Every exact computation in this unit can be checked with an estimate first. The The standard multiplication algorithm produces a product digit by digit, and a compatible estimate says how many digits it should have. In long division and in dividing by a two-digit number, each digit of the quotient is itself a small estimate, found by the same move as Example 2: make the divisor friendly and ask which basic fact fits. Benchmarks for fractions return whenever fractions are added or multiplied, as a check that an exact answer is the right size. Later, estimates are chained together to size up enormous quantities in Estimating a huge quantity from small facts, and the question of how far off a number may be comes back for measured quantities in Significant figures.

Exercises

Part 1: compatible numbers for division

1. Estimate 3{,}452 \div 6 using compatible numbers.
2. Estimate 2{,}781 \div 4 using compatible numbers.
3. Estimate 5{,}530 \div 9 using compatible numbers.
4. Estimate 1{,}912 \div 3 using compatible numbers.
5. Estimate 47{,}960 \div 8 using compatible numbers.
6. Estimate 2{,}054 \div 52 by making the divisor friendly first.

Part 1: at the food drive

7. Estimate 51 \times 19 using compatible numbers.
8. The school collected 3{,}215 cans and splits them equally among 8 food pantries. Estimate the number of cans for each pantry.
9. Satoshi wants to estimate 2{,}893 \div 7. He rounds to the nearest hundred and gets stuck. Why does rounding not help, and what compatible numbers give an estimate?

Part 2: benchmark fractions

10. Is each fraction near 0, near \tfrac{1}{2} or near 1: \tfrac{2}{9}, \tfrac{6}{11}, \tfrac{9}{10}, \tfrac{7}{15}?
11. Estimate \tfrac{11}{12} + \tfrac{3}{7} using benchmarks.
12. Estimate \tfrac{9}{10} - \tfrac{4}{9} using benchmarks.
13. Nayeli’s three boxes of pasta weigh 2\tfrac{7}{8}, 1\tfrac{1}{10} and 3\tfrac{4}{9} pounds. Estimate the total weight.

Part 2: benchmark decimals

14. Estimate 3.96 + 7.51 + 0.48.
15. Estimate 6.02 - 2.49.
16. Estimate 0.51 \times 64.
17. A class brought 72 cans, and \tfrac{4}{9} of them are soup. Estimate the number of soup cans using a benchmark, and say whether the estimate is high or low.

Part 3: over or under

18. Say whether each estimate is an overestimate, an underestimate, or cannot be told from the rounding alone. (a) 47 \times 68 \approx 50 \times 70. (b) 6{,}320 \div 9 \approx 6{,}300 \div 9. (c) 29 \times 61 \approx 30 \times 60.
19. Nayeli has 40 dollars for supplies costing 12.35 dollars, 9.80 dollars and 16.45 dollars. Use an estimate to decide whether 40 dollars is enough.
20. The gym has 14 rows of 32 seats. Satoshi needs seats for 400 people at the food drive assembly. Should he overestimate or underestimate the number of seats? Make that estimate and decide.
21. A truck can carry 2{,}500 pounds. The food drive has 96 boxes, each weighing 23 pounds. Use an estimate to decide whether one truck can carry them all.
22. Wenjing estimates 5{,}618 \div 8 \approx 5{,}600 \div 8 = 700. Without dividing 5{,}618 by 8, say whether 700 is an overestimate or an underestimate, and by exactly how much.
23. Nayeli rounds the can counts of 6 classes to the nearest ten and adds the rounded counts. Before seeing the counts, what is the most her total can be off by?

Reasoning

24. Explain, using cans shared among trucks, why making the divisor bigger makes a quotient estimate smaller.
25. Explain why rounding each price to the nearest dollar can fail to answer “is there enough money?” Give two prices and a budget where it fails.