The packing house fills crates of apples a day. About how many crates is that in days, and how can you tell in a few seconds whether an exact answer is the right size?

Grade 4 · 3.OA.D.8 · 7.EE.B.3

Estimating by rounding

The packing house fills 487 crates of apples a day. About how many crates is that in 6 days, and how can you tell in a few seconds whether an exact answer is the right size?

You need

Multi-digit multiplicationGrades 4 to 5 RoundingGrades 3 to 4 Rounding to any place, including decimalsGrade 4 The place value chartGrade 2 +1 more

Unlocks

Friendly numbers and deliberate over and under estimatesGrade 5

Where you are 5 in Estimation and mental math

Number9

Arithmetic5

Estimation and mental math5

Estimating sums and differences Mental strategies2

Estimating by rounding

Friendly numbers and deliberate over and under estimates Estimating a huge quantity from small facts

When you reach for it

Jatin and Mere work at an apple packing house. Apples come in from the orchards, get packed into crates, and leave on trucks. Their friend Necalli runs the ticket window at an ice skating rink. All three need answers like “about how many crates this week?” and “about how much money for six tickets?” An exact answer takes paper and time. Often “about” is enough.

There is a second reason to estimate. When you work out an exact answer, with paper or with a calculator, a quick estimate says what size the answer should be. If the two are close, the exact answer has the right size. If they are far apart, it is worth a second look.

The tool is one you already have: Rounding. To estimate, round each number first, then compute with the rounded numbers. How to round to a chosen place is in Rounding to any place, including decimals; this lesson is about using it.

The procedure

Part 1: round first, then compute

The packing house fills 487 crates of apples a day. About how many crates is that in 6 days?

The exact answer is 487 \times 6. That needs Multi-digit multiplication. But 487 is close to 500, and 500 \times 6 is easy: 5 \times 6 = 30, so 5 hundreds times 6 is 30 hundreds.

487 \times 6 \approx 500 \times 6 = 3{,}000

The sign \approx is read “is about.” It says the two sides are close, not equal. The packing house fills about 3{,}000 crates in 6 days. The exact number is 2{,}922, and 3{,}000 is close to it.

Why does this work? Rounding moves each number only a little. When the numbers move a little, their sum, difference or product also moves only a little. So the answer with rounded numbers lands near the real answer, and the rounded numbers are easy to work with.

Which place to round to. That depends on what the estimate is for.

For a sum or a difference, round every number to the same place, often the hundreds or the thousands. Then every rounded number ends in the same zeros, and adding them is adding a few digits in one column of the The place value chart.
For a product, a common choice is to round each factor to its leading place, its first digit from the left. Then each factor is one digit followed by zeros, and the product is a basic fact followed by zeros, as in Multiplying by tens and hundreds. For 487 \times 6, the 6 already has one digit, so it stays.

A finer place gives a closer estimate and a little more work. A coarser place gives a rougher estimate and less work. There is no single right place; there is a place that suits the question.

Front-end estimation is a faster cousin of rounding. Keep the leading digit of each number and write zeros in every place after it. The front end of 487 is 400, and the front end of 2{,}315 is 2{,}000. It is quick because there is nothing to decide: you never look at the next digit.

Figure 1487 lies past the halfway mark, so it rounds to 500. Front-end estimation keeps only the leading digit and gives 400, which is always at or below the number.

The number line shows the cost. Rounding sends 487 to 500, its nearer hundred. The front end drops the 87 and lands on 400, much farther away. So

487 \times 6 \approx 400 \times 6 = 2{,}400

is a rougher estimate than 3{,}000. A front end never moves a number up, so a front-end estimate of a sum or a product of whole numbers is never above the exact answer. For a sum there is a quick repair: after adding the front ends, estimate what was dropped and add that too (Example 4).

Part 2: is the answer reasonable?

An estimate is a check on an exact answer. Mere uses the The standard addition algorithm, the The standard subtraction algorithm or Partial products to get the exact answer, and an estimate to see that it has the right size.

An exact answer is reasonable when it is close to the estimate. “Close” means close for the size of the numbers. For 487 \times 6, the exact 2{,}922 is 78 away from the estimate 3{,}000, which is small next to 3{,}000. An answer such as 29{,}220 or 292 would be a whole place away from the estimate, ten times too big or ten times too small.

Which way the estimate is off. Sometimes you can tell without the exact answer.

If every number was rounded up (a number that is already round may stay as it is), the estimate of a sum or a product is too high. Bigger numbers make a bigger total and a bigger product.
If every number was rounded down (or stayed), the estimate of a sum or a product is too low.

For 487 \times 6, the 487 went up to 500 and the 6 stayed. So 3{,}000 must be a little more than the exact product. The area model shows why.

Figure 2Rounding 487 up to 500 adds the thin strip. The rounded product covers the exact one and a little more, so 3{,}000 is above 2{,}922.

The rectangle 500 wide and 6 tall holds the whole rectangle 487 wide and 6 tall, plus a thin strip 13 wide. Covering more means a bigger product. When one number goes up and another goes down, these moves alone do not tell which way the estimate is off. That case, and how far off an estimate can be, are in Friendly numbers and deliberate over and under estimates.

Decimals, such as money. The same move works with decimals. Round each price to the nearest dollar, which is rounding a decimal to the ones place, and then compute with whole numbers. 3.85 dollars is about 4 dollars, because 0.85 is past the halfway mark of 0.50. 12.20 dollars is about 12 dollars.

Worked examples

Examples 1 to 4 belong to Part 1. Examples 5 to 7 belong to Part 2.

Example 1

Crates on three trucks

On Monday three trucks leave the packing house with 318, 274 and 409 crates. About how many crates left in all?

Solution

Round each number to the nearest hundred. 318 is closer to 300 than to 400. 274 is past the halfway mark 250, so it is closer to 300. 409 is closer to 400.

318 + 274 + 409 \approx 300 + 300 + 400 = 1{,}000

Adding hundreds is adding the digits 3 + 3 + 4 = 10, and 10 hundreds is 1{,}000. About 1{,}000 crates left on Monday. (The exact total is 1{,}001.)

Example 2

How many more tickets

Necalli’s rink sold 2{,}846 tickets in December and 1{,}375 in November. About how many more tickets were sold in December? Estimate twice, rounding to the nearest hundred and to the nearest thousand.

Solution

Nearest hundred: 2{,}846 becomes 2{,}800 and 1{,}375 becomes 1{,}400.

2{,}846 - 1{,}375 \approx 2{,}800 - 1{,}400 = 1{,}400

That is 28 hundreds take away 14 hundreds, which leaves 14 hundreds.

Nearest thousand: 2{,}846 becomes 3{,}000 and 1{,}375 becomes 1{,}000.

2{,}846 - 1{,}375 \approx 3{,}000 - 1{,}000 = 2{,}000

The exact difference is 1{,}471. The hundreds estimate is off by 71; the thousands estimate is off by 529. To the thousands, the first number moved up and the second moved down, so the gap was stretched from both ends. Rounding to the thousands is quicker, but for this question the hundreds give an answer Necalli can use: about 1{,}400 more tickets.

Example 3

Apples on a truck

A crate holds 72 apples, and a truck holds 38 crates. About how many apples are on a full truck?

Solution

Round each factor to its leading place, the tens: 72 becomes 70 and 38 becomes 40.

72 \times 38 \approx 70 \times 40 = 2{,}800

To find 70 \times 40, multiply the leading digits, 7 \times 4 = 28. Each factor brought one zero, so the product has two: 7 tens times 4 tens is 28 hundreds, which is 2{,}800. A full truck carries about 2{,}800 apples. (The exact product is 2{,}736.)

Example 4

Front end, then a repair

Necalli adds up three weeks of ticket sales: 1{,}642, 2{,}315 and 1{,}908. Make a front-end estimate, then repair it.

Solution

Keep the leading digit of each number, the thousands:

1{,}000 + 2{,}000 + 1{,}000 = 4{,}000

That is too low, because the front end dropped 642, 315 and 908. Estimate the dropped parts by their hundreds and add them:

600 + 300 + 900 = 1{,}800

The repaired estimate is 4{,}000 + 1{,}800 = 5{,}800 tickets. The exact total is 5{,}865. The plain front end was off by more than 1{,}800; the repaired one is off by 65. Rounding each number to the nearest hundred gives 1{,}600 + 2{,}300 + 1{,}900 = 5{,}800 as well.

Example 5

Checking an exact product

The packing house sends out 23 truckloads of 168 crates each. Mere works out the product with partial products and gets 23 \times 168 = 3{,}864. Use an estimate to judge whether that answer is reasonable.

Solution

Round each factor to its leading place: 23 becomes 20 and 168 becomes 200.

23 \times 168 \approx 20 \times 200 = 4{,}000

since 2 \times 2 = 4 and the factors bring three zeros between them. Mere’s 3{,}864 is close to 4{,}000, so it has the right size. It is reasonable.

The estimate rules out answers like 38{,}640 or 386, which are a whole place away. It cannot tell 3{,}864 from 3{,}874, though. An estimate checks the size of an answer, not every digit. Here the exact answer is right: 20 \times 168 = 3{,}360 and 3 \times 168 = 504, and 3{,}360 + 504 = 3{,}864.

Example 6

Both numbers rounded up

A festival at the rink brings 46 school groups of 28 skaters each. Estimate the number of skaters. Without finding the exact product, say whether it is more or less than the estimate.

Solution

Round each factor to the nearest ten: 46 becomes 50 and 28 becomes 30.

46 \times 28 \approx 50 \times 30 = 1{,}500

Both factors moved up. A rectangle 50 by 30 holds a rectangle 46 by 28 with room to spare, so the exact product is less than 1{,}500. About 1{,}500 skaters, and certainly fewer. (The exact number is 1{,}288.)

Example 7

Money to the nearest dollar

(a) Necalli buys a skate rental for 4.75 dollars, a ticket for 8.50 dollars and a hot cocoa for 2.35 dollars. Estimate the total. (b) Jatin buys 6 tickets at 8.95 dollars each. Estimate the cost, and say whether the exact cost is more or less.

Solution

(a) Round each price to the nearest dollar. 4.75 is past 4.50, so it becomes 5. 8.50 sits exactly on the halfway mark. “Nearest” does not decide, so the usual convention from Rounding to any place, including decimals applies: a number on the halfway mark rounds up, to 9. 2.35 is below 2.50, so it becomes 2.

4.75 + 8.50 + 2.35 \approx 5 + 9 + 2 = 16

About 16 dollars. (The exact total is 15.60 dollars.)

(b) 8.95 is just under 9.

6 \times 8.95 \approx 6 \times 9 = 54

About 54 dollars. The price was rounded up and the 6 stayed, so the exact cost is a little less than 54 dollars. (It is 53.70 dollars.)

Boundaries

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 its size.
An estimate checks size, not digits. It catches an answer a whole place off, but two answers that differ by a few ones look the same to it (Example 5).
Rounding to one place can lose a small difference. 4{,}512 - 4{,}489 \approx 4{,}500 - 4{,}500 = 0, yet the exact difference is 23. When two numbers are close together, round to a finer place or subtract exactly.
In a product, a rounding move is multiplied. Moving 3{,}420 to 3{,}000 moves it by 420, and times 5 that is a move of 2{,}100. A coarse place in a product can give a poor estimate.
The “too high” and “too low” rules are for sums and products when every number moved the same way. They do not hold for a difference, where raising the number taken away makes the result smaller. Both numbers go up in 58 - 26 \approx 60 - 30 = 30, yet the estimate is below the exact 32. When numbers move in different directions in a product, the rules say nothing.

Where this goes next

Every exact computation from here on can start with an estimate. The The standard multiplication algorithm builds a product digit by digit, and an estimate says how many digits the answer should have. When plain rounding leaves numbers that are still hard to multiply or divide, Friendly numbers and deliberate over and under estimates picks nearby numbers that fit a basic fact, and also measures how far off an estimate is. Rounding prices to the nearest dollar comes back in Multiplying decimals, where an estimate says where the decimal point belongs. With fractions, the same idea rounds to 0, \tfrac{1}{2} or 1 in Estimating with fractions.

Exercises

Part 1: sums and differences

Estimate 392 + 517 by rounding to the nearest hundred.
Estimate 1{,}286 + 3{,}719 by rounding to the nearest hundred.
Estimate 6{,}342 - 2{,}781 by rounding to the nearest hundred.
Estimate 8{,}904 - 4{,}156 by rounding to the nearest thousand.
The packing house filled 624 crates on Monday, 587 on Tuesday and 451 on Wednesday. Estimate the total by rounding to the nearest hundred.

Part 1: products

Estimate 8 \times 693 by rounding 693 to its leading place.
Estimate 4 \times 2{,}385 twice: rounding 2{,}385 to the nearest thousand, then to the nearest hundred. The exact product is 9{,}540. Which estimate is closer?
Estimate 53 \times 68.
Estimate 29 \times 81.
The rink runs 7 sessions a week, and 418 skaters come to each one. Estimate the number of skaters in a week.
Each crate holds 48 apples. An order is 215 crates. Estimate the number of apples in the order.

Part 1: front-end estimation

Make a front-end estimate of 4{,}725 + 3{,}190, then repair it using the dropped hundreds.
Make a front-end estimate of 6 \times 573. Is it above or below the exact product?
Jatin adds three loads: 2{,}618, 1{,}457 and 3{,}204 apples. Make a front-end estimate, then repair it using the dropped hundreds.

Part 2: is it reasonable?

Jatin works out 7 \times 386 = 2{,}702. Use an estimate to judge whether that is reasonable.
Necalli’s calculator shows 62 \times 384 = 23{,}808. Is that reasonable?
One of these is 9 \times 4{,}108: 3{,}697, 36{,}972 or 369{,}720. Use an estimate to pick it.
The packing house shipped 5{,}214 crates in May and 2{,}968 in June. Mere finds that May had 2{,}246 more. Check her answer with an estimate.

Part 2: which way, and money

Estimate 37 \times 58 by rounding each factor to the nearest ten. Without multiplying exactly, is the exact product more or less than your estimate?
Estimate 4 \times 1{,}215 by rounding 1{,}215 to the nearest hundred. Is the exact product more or less than your estimate?
The rink’s two Saturday sessions sold 386 and 279 tickets. Estimate the total to the nearest hundred. Without adding exactly, is the exact total more or less than your estimate?
Estimate 3.79 + 12.25 + 6.60 dollars by rounding to the nearest dollar.
A class rents 8 pairs of skates at 4.85 dollars a pair. Estimate the cost, and say whether the exact cost is more or less than the estimate.

Reasoning

Explain why a front-end estimate of a sum of whole numbers is never more than the exact sum.
Explain why an estimate can show that an answer is the wrong size but cannot show that every digit is right. Use 7 \times 386 as your example.

Answers

About 900. 400 + 500 = 900. (Exact: 909.)
About 5{,}000. 1{,}300 + 3{,}700 = 5{,}000. (Exact: 5{,}005.)
About 3{,}500. 6{,}300 - 2{,}800 = 3{,}500. (Exact: 3{,}561.)
About 5{,}000. 9{,}000 - 4{,}000 = 5{,}000. (Exact: 4{,}748.)
About 1{,}700 crates. 600 + 600 + 500 = 1{,}700. (Exact: 1{,}662.)
About 5{,}600. 8 \times 700 = 5{,}600, since 8 \times 7 = 56. (Exact: 5{,}544.)
4 \times 2{,}000 = 8{,}000 and 4 \times 2{,}400 = 9{,}600. The hundreds estimate is closer: it is off by 60, while the thousands estimate is off by 1{,}540.
About 3{,}500. 50 \times 70 = 3{,}500. (Exact: 3{,}604.)
About 2{,}400. 30 \times 80 = 2{,}400. (Exact: 2{,}349.)
About 2{,}800 skaters. 7 \times 400 = 2{,}800. (Exact: 2{,}926.)
About 10{,}000 apples. 50 \times 200 = 10{,}000, since 5 \times 2 = 10 and the factors bring three zeros. (Exact: 10{,}320.)
Front end: 4{,}000 + 3{,}000 = 7{,}000. Repair: the dropped parts are about 700 + 100 = 800, so about 7{,}800. (Exact: 7{,}915.)
6 \times 500 = 3{,}000. It is below the exact product, because the front end moved 573 down. (Exact: 3{,}438.)
Front end: 2{,}000 + 1{,}000 + 3{,}000 = 6{,}000. Repair: 600 + 400 + 200 = 1{,}200, so about 7{,}200 apples. (Exact: 7{,}279.)
Reasonable. 7 \times 400 = 2{,}800, and 2{,}702 is close to it.
Reasonable. 60 \times 400 = 24{,}000, and 23{,}808 is close to it.
36{,}972. 9 \times 4{,}000 = 36{,}000, and only 36{,}972 is near that; the other two are a whole place away.
Reasonable. 5{,}200 - 3{,}000 = 2{,}200, close to 2{,}246.
40 \times 60 = 2{,}400. The exact product is less, because both factors were rounded up. (Exact: 2{,}146.)
4 \times 1{,}200 = 4{,}800. The exact product is more, because 1{,}215 was rounded down and the 4 stayed. (Exact: 4{,}860.)
400 + 300 = 700. The exact total is less, because both numbers were rounded up. (Exact: 665.)
About 23 dollars. 4 + 12 + 7 = 23. (Exact: 22.64 dollars.)
About 40 dollars. 8 \times 5 = 40. The exact cost is less, because 4.85 was rounded up. (Exact: 38.80 dollars.)
The front end of a whole number keeps its leading digit and replaces every later digit with 0, so it is the number itself or smaller. Adding numbers that are each at most the real ones gives a total that is at most the real total.
The estimate 7 \times 400 = 2{,}800 says the product is in the thousands, a little under 2{,}800. An answer like 27{,}020 or 270 is far from it, so it is the wrong size. But 2{,}702 and 2{,}712 are both close to 2{,}800; the estimate cannot tell which is right. Only the exact work, or a check such as 7 \times 386 = 2{,}100 + 560 + 42 = 2{,}702, settles the digits.