# Estimation in multiplication

This is a complete lesson with instruction and exercises about using estimation in multiplication, meant for 5th or 6th grade. To estimate, students round two and three-digit numbers before multiplying, but this rounding can be done in several different ways. Various exercises and word problems follow.

To estimate the result of multiplication (product), round the numbers to some close numbers that you can easily multiply mentally.

One method of estimation is to round all factors to the biggest digit (place value) they have.
(This is somewhat of a crude method but serves as a starting point in learning estimation.)

For example, estimate  365 × 24.  Round 365 to the nearest hundred, and 24 to the nearest ten.  So 365 ≈ 400, and 24 ≈ 20.  Then  365 × 24 ≈ 400 × 20 = 8000.  This way the multiplication is easy to do since it is only a matter of a single digit (4) times a single digit (2), and tagging zeros to the end (000).

Look at other examples:

Estimation

 133 × 27 ≈  100 × 30 = 3000

In reality:

 2 2   133  ×   27 931 2660 3591

Estimation

 79 × 73 ≈ 80 × 70 = 5600

In reality:

 6 2   79  ×   73 237 5530 5767

## Practice

1.  Estimate the products by rounding the factors to the biggest place value.

 a.    158 × 32 ≈ 200 × 30 = 6000 c.   29 × 94 ≈ ____ × ___ = ____ d.    770 × 33 ≈ ____ × ___ = ____ f.  88 × 99 ≈ ____ × ___ = ____ j.     486 × 21 ≈ ____ × ___ = ____ l.   209 × 27 ≈ ____ × ___ = ____

2.  One purpose of estimation is to catch gross errors in calculations.  For example, if you estimate the result to be 5000, and you calculate it to be 354, you know something is wrong since you're way off.
What is best estimate of the options given?

 1.    103 × 52 2.    42 × 76 3.    319 × 25 4.    17 × 17 5.    99 × 59 6.    47 × 21 a.  6500 a.  4000 a.  6000 a.  1000 a.  6000 a.  470 b.  500 b.  320 b.  750 b.  200 b.  900 b.  9700 c.  5000 c.  4800 c.  9000 c.  400 c.  9000 c.  1000

3.  Which product is the furthest from its estimate?  Can you see why?

 Estimations:236 × 28 ≈ 198 × 28 ≈ 246 × 28 ≈ 178 × 28 ≈ Exact products:236 × 28 = 198 × 28 = 246 × 28 = 178 × 28 =

 Other methods of estimation When you round both factors to the biggest place value, 249 × 34 would be estimated to 6000; yet the actual product is 8466 - quite far from the estimate.   You get a much better estimate if you round 249 to 250, and not to 200.  Round 34 still to 30.  So 249 × 34 ≈ 250 × 30.  To calculate this, first go 3 × 25 = 25 + 25 + 25 = 75, and then tag two zeros to it: 250 × 30 = 30 × 250 = 7500. So instead of rounding to nearest hundred, you can round to the 'middle fifty' if the number is close to it. Or, even when the number has hundreds, you can round to nearest ten. For example, 178 × 28 is better estimated to 180 × 30 - rounding both numbers to nearest ten.  That you can do mentally by going 100 × 30 + 80 × 30, which is 3000 + 2400 = 5400. You can also try to account for the error in your estimation by rounding one factor up, the other down - especially if you have to round one factor a lot.  For example, 144 × 24.  If you round 144 down to 100, round 24 up to 30 to make up for some of the error.  Estimation is not an exact science but a matter of rounding to close numbers that you can work in your head.

4. Estimate first, and then calculate the actual product. You can use various ways to estimate as you see fit.  Or, try two different methods of estimation and compare which was more accurate.

 Estimations: Reality: a. 143 × 27 ≈  ________ × _________ = __________ 143 × 27 = _________ b. 657 × 13 ≈  ________ × _________ = __________ 657 × 13 = _________ c. 411 ×  9  ≈  ________ × _________ = __________ 411 ×   9  = _________ i. 406 × 19 ≈  ________ × _________ = __________ 406 × 19 = _________ k. 243 × 24 ≈  ________ × _________ = __________ 243 × 24 = _________ l. 37 × 258 ≈  ________ × _________ = __________ 37 × 258 = _________

5.  Solve the word problems. If necessary, round the numbers to make an estimate.

a.  Nelly counted the matches in a match box and got 58.
How many matches approximately are in 8 boxes?

Each day Nelly uses 10 matches.  How long approximately will those 8 boxes last her?

b.  Aunt Jeanie's chickens lay about 130 eggs a day.
Can she supply all the eggs for the above school within one week?
Note, her chickens lay eggs 7 days a week.

c.  One (food) can costs 58 cents, and you're going to buy 18 of them.
Make an estimation of the cost.

Based on your estimation, will \$10 be enough money for your purchase?