ABE Math Tutorials/Whole numbers/Estimation

Estimation

Putting it Together

Reflection

Do you remember the lesson on "Rounding"? (Click here if you don't.) In that lesson, we learned to round numbers to the nearest hundred, thousand, or whatever was requested. But in "real life", it's rare that we're asked to round single numbers. More often, we're trying to round 2 or 3 numbers so that we can calculate an answer in a hurry. For example, we often need to calculate the total price:

Activity

Marta buys 2 loaves of bread at 97¢ each, and a carton of milk for $2.12. Will she be able to pay for these groceries with a five-dollar bill?

Actual Value	Approximation	Estimated value
$39.42	rounds up to	$40
$46.91	rounds up to	$50
$48.55	rounds up to	$50

• To find the answer to this problem, we will probably choose to round all the numbers to the nearest dollar:

Bread (2 loaves) 97¢ rounds to: $1.00 97¢ rounds to: $1.00 Milk $2.12 rounds to: $2.00

   Total: about $4.00 

• • Answer: Yes, Marta will be able to pay for these groceries with a five-dollar bill.

Here's another example:

Suppose that you need 2 new speakers for your stereo system. Speakers are on sale for $83 each. How much can you expect to pay for the 2 speakers? 

To answer this question, you might first want to round the price to the nearest ten dollars, giving a rounded speaker price of $80. Then you could just multiply 80 x 2 (which is just 8 x 2, with a zero added on to the answer):

80 x 2 = $160

! However, in "real life", you may live in a place where a sales tax is added to most of the things that you buy. In this case, you would probably rather round the price up instead. In this way, you will have a more realistic idea of how much you can expect to pay at the checkout: $83 rounds up to $90 90 x 2 = $180

Answer:You can expect to pay about $180 for the speakers.

Here's another example where rounding combines with powers of 10 to make our arithmetic easier: Julia won $649 at a Bingo game. She has decided to divide the amount evenly between her 8 grandchildren. About how much will each child receive?

We can see right away that this is going to be a division problem: the exact answer will be 649 ÷ 8. But to quickly figure out about how much each child will receive, let's round the big number to a number which will divide easily by 8. If you think about the 8 times tables, you'll remember that 8 x 8 = 64. Let's round the 649 to 640 to make our division easier:

640 ÷ 8 = 80

Each child will receive about $80. 

Here's an example with more than 2 numbers: Kaitlin often buys propane for her camper. Here is a table of what she paid for propane in the four seasons last year:

season: propane bill: Spring $39.42 Summer $46.91 Fall $48.55 Winter $35.08

About how much did Kaitlin spend on propane for the year?

To answer this question, you would probably begin by rounding each of the bill amounts to the nearest ten-dollar amount, like this:

$39.42...... rounds to: $40 $46.91...... rounds to: $50 $48.55...... rounds to: $50 $35.08...... rounds to: $40

But let's look at these numbers a little more closely. We had 4 numbers to start with, and we ended up rounding all of them up. We will probably end up with a total which is quite a bit higher than the real total of these 4 amounts. We will get a more accurate picture of the true total if we take the $35.08 value, and round it down to $30. Now our rounding table looks like this:

$39.42...... rounds to: $40 $46.91...... rounds to: $50 $48.55...... rounds to: $50 $35.08...... rounds to: $30 Total: About $170

Summary

Why??? The process of rounding large numbers and then doing a "quick arithmetic" with powers of ten, is called estimation. Why do we bother to estimate? Why not just add, subtract, multiply or divide numbers just as they are? Here are some easy reasons: Rounding numbers, so that we can do the arithmetic operation in our heads, is so much quicker than "long arithmetic". Estimating gives us a realistic idea of what the answer should be. Estimation gives us an answer which is often "good enough". For example, we usually don't need to know exactly what our monthly expenses will be. If we have a rough number, we can budget. Many answers on the GED math test can be estimated well enough without doing the long arithmetic!

Here's one more example: Sher has offered to provide some jars for her community's Bake Sale. She knows that she has 18 boxes full of jars, and that each box holds 57 jars. About how many jars should she offer to give?

This is obviously a multiplication problem: we know that we should be multiplying 18 x 57 to find the total number of jars. When rounding these numbers, you might be tempted to just round the 18 up to 20; then round the 57 up to 60. But this is another situation where we must be realistic about our estimating. If we round both numbers up, we will end up with a very large number -- Sher might end up promising many more jars than she actually has! (And, in fact, some of the jars may even be broken.) In a case like this one, it makes sense to round the 18 up to 20; but to round the 57 down to 50 to do our calculations:

Answer: She should offer to give about 20 x 50 = 1,000 jars to the Bake Sale. Ready to try some estimation problems on your own?

Estimate the following amounts: