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Estimation Of Results Of Arithmetic Operations

Estimating Results of Arithmetic Operations of Whole Numbers

  • When we perform any arithmetic operation on the whole numbers and we do not need an exact answer, then we can round off the whole numbers.
  • The arithmetic operations are addition, subtraction, multiplication and division.
  • Before performing any arithmetic operations on the whole numbers we round off them and then we can perform these operations.

 Sum is the answer to an addition problem.

 Difference is the answer of a subtraction problem.

  • A product is an answer of a multiplication problem.
  • A quotient is an answer of a division problem.
  • To estimate sums, differences, products and quotients, we can round off the numbers that we are working with, to the nearest power of ten.

Illustration Questions

Estimate the value of \(824+166\).

A \(1000\)

B \(1100\)

C \(900\)

D \(1200\)

×

Analyze \(824\)

The digit at the tens place is less than \(5.\)

So, \(824\) is rounded down to the nearest hundred place i.e. \(800\).

Analyze \(166\)

The digit at the tens place is greater than \(5\).

So, \(166\) is rounded up to the nearest hundred place i.e. \(200\).

Now adding the rounded values,

\(800+200=1000\)

Hence, option (A) is correct.

Estimate the value of \(824+166\).

A

\(1000\)

.

B

\(1100\)

C

\(900\)

D

\(1200\)

Option A is Correct

Estimating Results of Arithmetic Operations of Decimal Numbers

  • To perform any arithmetic operation on decimal numbers, first, we round off them and then perform the operations.
  • We can round off the decimals to the nearest whole number and then we can perform the arithmetic operations such as addition, subtraction, multiplication and division to find a reasonable solution of our problem.
  • To find an estimated answer of sums, differences, products and quotients, we use the rounding rules.

Illustration Questions

What is the estimated value of \(18.627+20.231\)?

A \(40\)

B \(37\)

C \(38\)

D \(39\)

×

Analyze \(18.627\)

The digit at tenths place is \(6\) which is greater than \(5.\)

So, the digit at the ones place is increased by \(1\) and the digits after decimal point become zero, i.e. \(19\).

Analyze \(20.231\)

The digit at the tenths place is less than \(5.\)

So, the digit at the ones place remains same and the digits after decimal point become zero, i.e. \(20\).

Now adding the rounded values,

\(19+20=39\)

Hence, option (D) is correct.

What is the estimated value of \(18.627+20.231\)?

A

\(40\)

.

B

\(37\)

C

\(38\)

D

\(39\)

Option D is Correct

Estimating Results of Arithmetic Operations of Fractions

  • To perform any operation on fractions, first, we round off them to the nearest half and then perform the operation.
  • To estimate the sums, differences, products and quotients of the simple fractions or mixed fractions, we use the rounding rules of simple fraction or mixed fraction that we discussed earlier.

We can round off a simple fraction to the nearest half which can be \(0,\;\dfrac{1}{2}\) or \(1.\)

We can round off a mixed fraction to the nearest whole number.

Illustration Questions

Estimate the value of \(5\dfrac{1}{3}+\dfrac{1}{8}\).

A \(5\)

B \(0\)

C \(1\)

D \(6\)

×

Consider \(5\dfrac{1}{3}\).

\(5\dfrac{1}{3}\) lies between \(5\) and \(6\) and the nearest half of \(\dfrac{1}{3}\) is zero.

So, mixed fraction \(5\dfrac{1}{3}\) is rounded to the whole number \(5.\)

Consider \(\dfrac{1}{8}\).

\(\dfrac{1}{8}\) is closest to zero.

So, \(\dfrac{1}{8}\) is rounded to zero.

Now rewriting the problem with rounded values,

\(5+0=5\)

Hence, option (A) is correct.

Estimate the value of \(5\dfrac{1}{3}+\dfrac{1}{8}\).

A

\(5\)

.

B

\(0\)

C

\(1\)

D

\(6\)

Option A is Correct

Front-End Estimation of Whole Numbers

  • Front-End Estimation method is a useful method of estimating when adding and subtracting the numbers greater than \(1000\).

Steps for Front-End Estimation Method

  • When number is greater than \(1000\), front end estimation method is applicable.
  1. Retain digits of the two highest place values in the number.
  2. Put zeros on other place values.

Example: \(5324+5683\)

  • Each number is greater than \(1000\) so, Front-End Estimation method is applicable.
  • Consider \(5324\).

Keep \(5\) and \(3\) which are the digits of two highest place values and insert zero for the other place values so,

\(5324\) becomes \(5300\).

Consider \(5683\).

Keep \(5\) and \(6\) which are the digits of two highest place values and insert zeros for the other place values so, \(5683\) becomes \(5600\).

Rewrite the problem and perform the operation.

\(5300+5600=10900\)

Illustration Questions

Estimate the value of \(1264-1075\) by using Front-End estimation method.

A \(100\)

B \(300\)

C \(200\)

D \(400\)

×

Consider \(1264\).

Keep digits of the two highest place value i.e. \(1\) and \(2\) and put zeros for other place values.

Now \(1264\) becomes \(1200\).

Consider \(1075\).

Keep digits of the two highest place value i.e. \(1\) and \(0\) and put zeros for other place values.

Now \(1075\) becomes \(1000\).

Now rewriting the problem with rounded values,

\(1200-1000=200\)

Hence, option (C) is correct.

Estimate the value of \(1264-1075\) by using Front-End estimation method.

A

\(100\)

.

B

\(300\)

C

\(200\)

D

\(400\)

Option C is Correct

Front-End Estimation of Decimal Number

  • While using the Front-End Estimation method with decimals, we separate the whole number and the decimal part from decimal number and then combine them together after performing operations.

Steps for Front-End Estimation Method for Decimal Numbers

\(\to\) Write the Front digits of the numbers being added or subtracted and perform operation.

\(\to\) Round off the decimal part of the numbers being added or subtracted and then perform operation.

\(\to\) Then combine the result.

Example: \(5.20+9.66\)

\(\to\) Front digits of the numbers being added are \(5\) and \(9\).

so, \(5+9=14\)

\(\to\) Round off the decimal parts.

\(0.20\) becomes \(0.2\) (no change).

\(0.66\) becomes \(0.7\).

So, \(0.2+0.7=0.90\)

Combine the result,

i.e. \(14+0.90=14.90\)

Illustration Questions

What is the estimated value of \(22.68-15.21\) by using Front-End Estimation method?

A \(7.50\)

B \(8.00\)

C \(8.50\)

D \(7.00\)

×

Front digits of the numbers being subtracted are \(22\) and \(15\).

So, \(22-15=7\)

Now round off the decimal parts.

\(0.68\) becomes \(0.70\) and \(0.21\) becomes \(0.20\).

So, \(0.70-0.20\)

\(=0.50\)

On combining the answer,

\(7+0.50=7.50\)

Hence, option (A) is correct.

What is the estimated value of \(22.68-15.21\) by using Front-End Estimation method?

A

\(7.50\)

.

B

\(8.00\)

C

\(8.50\)

D

\(7.00\)

Option A is Correct

Compatible Numbers

  • Compatible numbers are numbers that are close in value to the actual numbers and which make it easy to perform mental arithmetic.
  • The word compatible means "well-matched".
  • Compatible numbers are numbers that are friendly with each other.

For example: \(15\) and \(5\) are compatible numbers when it come to division.

  • Compatible numbers are useful in estimating the sum, difference, product or quotient.
  • Compatible number are easy to add, subtract, multiply or divide.
  • By rounding off the whole numbers, fractions or decimals, to the compatible numbers, we can easily perform the division operation.

Example: Estimate the value of \(29\div6.5\)

  • To estimate the quotient, firstly we round off the numbers \(29\) and \(6.5\).
  • \(6.5\) can be rounded to \(7\).
  • \(29\) can be rounded to \(30.\) But \(30\) is not exactly divisible by \(7\) so, we round down \(29\) to \(28\) because \(28\) and \(7\) are compatible numbers.
  • Now we can easily perform division.
  • Now rewrite the problem using rounded values, so \(28\div7=4\)

Illustration Questions

How many \(4\dfrac{7}{8}\) inch long pieces of ribbon can be cut from a \(20\dfrac{1}{2}\) inch long piece of ribbon? Give a reasonable estimate.

A \(2\)

B \(4\)

C \(5\)

D \(15\)

×

We have the numbers, \(4\dfrac{7}{8}\) and \(20\dfrac{1}{2}\).

We can round off the fraction \(4\dfrac{7}{8}\) to whole number \(5\) and fraction \(20\dfrac{1}{2}\) to whole number \(21\).

But \(21\) is not compatible number to \(5,\) so we round off it to \(20\).

Now \(20\) and \(5\) are compatible numbers.

 We can solve the problem as

\(20\div5=4\)

So we can cut \(4\) pieces of ribbon from a \(20\) inch long piece of ribbon.

Hence, option (B) is correct.

How many \(4\dfrac{7}{8}\) inch long pieces of ribbon can be cut from a \(20\dfrac{1}{2}\) inch long piece of ribbon? Give a reasonable estimate.

A

\(2\)

.

B

\(4\)

C

\(5\)

D

\(15\)

Option B is Correct

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