Estimating the Sums and Differences of Decimals

Introduction:

If you notice the answers above, they are not exact numbers, they are approximate numbers, or you’re guessing.

In mathematics, we often look for exact answers. But in real life, a little difference in the answer does not make any difference.

This guessing is known as estimation.

Estimation is nothing but finding a number that is close enough to the exact answer.

The sign for estimation is “ ”.

Let us consider the below situation.

“You want to arrange chairs in a row. The row is 23.6 m long, and the condition for arranging chairs is that the distance between each chair is 6 cm. How many chairs can be arranged in that row?”

Here, we are going to use rounding the numbers.

23.6 m is close to 24.

And the distance between each chair is 6 m.

Hence, approximately 4 (i.e, 24 6) chairs can be arranged in one row.

In this kind of situation, we can use the skill of estimation.

Why should we use estimation?

We should use estimation because,

• It is a skill for life
• It helps us to focus on
• It sometimes saves time
• It helps to gain confidence, helps in judgment and in decision making

 ESTIMATING SUMS AND DIFFERENCES OF DECIMALS

Let us look at another situation.

Tim went to a departmental store to purchase two shirts with the amount he had in his pocket. One shirt costs $49.50 and the other one $51.75. He wants to estimate his total before billing them. But he is not sure how to estimate the total.

Let us help him in estimating the total cost.

The easiest way to estimate a sum or a difference of decimals is to round the decimals.

If we round the decimals to the nearest whole numbers, we can use mental math to add or subtract decimals.

Let us estimate the cost of two shirts purchased by Tim.

$49.50 is close to $50

$ 51.75 is close to $52

Hence, two shirts may cost approximately $102.

There are two ways to find an estimate.

1. Round each addend to the nearest whole number:
   To round a decimal number to the nearest whole number, we need to check the digits after the decimal point. If the digit is less than 5, round the previous digit down; if it’s 5 or greater, round the previous digit up.
   Rounding the previous digit down means leaving the digit as it is, and rounding the previous digit up means increasing the digit by 1.
2. Substituting compatible digits:
   Compatible numbers are pairs of numbers that are easy to add, subtract mentally. When using estimation to approximate a calculation, replace actual numbers with compatible numbers.

Examples:

1. Estimate 15.7+4.9.
   15.7 à close to 16
   4.9 à close to 5
   à 16+5= 21
   The sum of 15.7+4.9 is approximately 21.

• Estimate 350.12+130.78.
  350.12 à close to 350
  130.78 à close to 131
  That is 350+131=481
  The sum of 350.12+130.78 is approximately 481.

• Estimate 45.78 – 22.33.
  We can also estimate by substituting compatible numbers.
  45.78 à 50
  22.33à 20
  50-20 = 30
  Hence, the sum of 45.78-22.33 is about 30.

• Estimate 86.9 – 18.
  Rounding the numbers to the nearest whole number.
  86.9 à close to 87
  18 is close to 20
  87 – 20 = 57
  Hence, the difference of 86.9 – 18 is approximately 57.

• Brenda and Jenna saved up $81.75 and $45.25, respectively to buy a gift for Mother’s Day. Estimate their total savings.
  $81.75+$45.25
  $81.75 is close to $82
  $ 45.25 is close to $45
  $82+$45 = $127
  They both together saved approximately $127.

Concept map:

Exercise:

• Estimate the sums and differences
• 56.32 + 23.12 = _____
• 18.76 + 11.23 = _____
• 14.56 + 76.98 = ______
• 67.19 – 33.12 = _____
• 88.92 – 33.10 = _____
• 76.56 – 3.45 = _____

• Three rock samples have masses of 74.05 g, 15.75 g and 9.01 g. A scientist estimates the total mass of the samples by rounding each mass to the nearest whole number. Which list includes the numbers he rounded up
• 74, 15, 9
• 75, 16, 9
• 74, 16, 9
• 74, 15, 10

What we have learned:

• Estimation is nothing but finding a number that is close enough to the exact answer.
• It is denoted by the sign “≈”.
• There is more than one way to estimate the sum or difference of decimals.
• Rounding each add end to the nearest whole number.
• Substituting compatible numbers.
• Estimation helps us to save time.
• Estimation helps us to gain confidence and helps in decision making.