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Mushadi Iksan, M.Ed.
CHAPTER ONE
ESTIMATION
ESTIMATION
Based on the Oxford Advanced Learner’s Dictionary there are two definitions of estimation.
The first definition of estimation is a judgement or opinion about the value or quality of somebody or something. For example, who is the best candidate in your estimation?
Meanwhile, the second definition of estimation is a judgement about the levels or quantity of something. For example, the estimation of our total sales is around 10 million.
Now study the following explanation.
How would listeners react if a football commentator announced that there were 48 271 people sitting in the stands at the Gelora Bung Karno waiting for the football match to begin? Does anyone care?
It is more usual to hear that there are 48 000 or 50 000 spectators as it is often not necessary to know the exact number of people. An estimate is enough, so the nearest rounded number is used.
We often need to use estimation when getting a precise answer is impossible, unnecessary, or inconvenient.
Estimation is not the same as a guess because it is based on information.
For example, we may know how many people are able to fit into the football ground and the approximate percentage of seats filled. We can use this information to produce our estimate.
In order to get the best estimation, there are some estimation strategies as follows:
• Estimate by Rounding
• Estimate by Clustering
• Estimate by Using Compatible Numbers
• Estimate by Using Front-End Estimation
• Rounding and Chopping
In this chapter we will discuss deeply about rounding and estimate by rounding. Meanwhile the others strategies will explain in brief as enrichment.
ROUNDING
In rounding strategies, we may round up, round down or round off to the nearest. For example, we round up Rp 85,000.00 to Rp 100,000.00 when we budget for a trip that will cost at least Rp 85,000.00. At NTUC supermarket, Singapore, the bill is round down to the nearest 5 cents. For example, if our bill is $ 12.03 then we pay $ 12.00.
In mathematics we round off the number to the nearest.
· Include one extra digit for consideration.
· Simply drop the extra digit if it is less than 5.
· If it is 5 or more, add 1 to the previous digit before dropping the extra digit.
Example 1
Round off the following number to (i) the nearest whole number, (ii) 2 decimal places, and (iii) 3 decimal places.
a. 9.7168 b. 19.2147 c. 0.82514
Solution:
a. (i) 9.7168 10 (ii) 9.7169 9.72 (iii) 9.7168 9.717
This digit is more than 5
This digit is more than 5
This digit is more than 5
b. (i) 19.2147 19 (ii) 19.2147 19.21 (iii) 19.2147 19.215
This digit is more than 5
This digit is less than 5
This digit is less than 5
c.
This digit is less than 5(i) 0.82514 1 (ii) 0.82514 0.83 (iii) 0.82514 0.825
This digit is more than 5
This digit is 5
Example 2
Round off the following number to (i) the nearest hundred and (ii) the nearest ten.
a. 179 b. 139 c. 124
Solution:
a. (i) 179 200 (ii) 179 180
b. (i) 139 100 (ii) 139 140
c. (i) 124 100 (ii) 124 120
EXERCISE 1
1. Write the following correct to (1) the nearest whole number and (ii) 2 decimal places:
a. 5.42467 b. 15.824 c. 7.862 d. 130.629
2. Write the following correct to 3 decimal places:
a. 712.8926 b. 0.00272 c. 0.8274 d. 7.024489
3. Express as decimal and give your answer correct to 3 decimal places.
4. Express the following fractions as decimal correct to 2 decimal places.
a. b. c. d.
5. Round off the following to (i) the nearest ten and (ii) the nearest hundred:
a. 7 029 c. 5 624 c. 8 790 d. 956
ESTIMATE BY ROUNDING
Example 1
Estimate the result of
a. 189.2 315.6
b.
Solution:
a. 189.2 x 315.6
Round off each number to the nearest hundred.
189.2 à 200
315.6 à 300
Then multiply 200 x 300
So, the product of 189.2 x 315.6 is about 60,000
b.
Round off each number to the nearest ten.
à 450
à 70
Then add 450 + 70
So, the sum of is about 520
EXERCISE 2
1. Estimate the answers to each of these:
a. 5961 + 1768
b. 432 – 192
c. 48 022 538
d. 9701 x 37
e. 98 631 + 608 897
f. 6501 + 3790
g. 11 890 – 3642
h. 83 481 1751
i. 112 000 x 83
j. 66 501 738
k. 392 x 113 486
l. 12 476 24
2. During a sale, one kilogram of fish was sold for £ 4.95. Estimate how many kilograms of fish you could buy with £20.
3. Without doing an exact calculation, determine whether you can afford all the items below if you have only £30.
· 1 two-kilogram bottle of corn oil for £6.95.
· 5 cans of peach at £1.95 per can.
· 300 g of beef at £1.02 per 100 g.
· 24 pockets of recombined milk at £2.85 for 6.
4. Estimate the value of 52.97603 – 31.32186
5. Estimate the value of
a. b.
THE OTHER ESTIMATE STRATEGIES
1. Estimate by Clustering
Example:
• 99.7 + 97.83 + 102.18 + 100.101 + 99.98
All of the numbers are close to 100.
There are five numbers.
The sum is about 5 x 100 = 500
•
All of the numbers are close to 15.
There are four numbers.
The sum is about 4 x 15 = 60
2. Estimate by Using Compatible Numbers
Example:
• 76.36 24.73
76.36 is close to 75.
24.73 is close to 25.
The quotient of 76.36 24.73 is about 75 25 = 3
•
is close to 7
is close to 21
The sum of is about 40
3. Estimate by Using Front-End Estimation
There are 4 techniques of estimate by using Front-End Estimation.
3.1 Range
Example:
• 257 + 576
Lowest range: 200 + 500 = 700
Highest range: 300 + 600 = 900
The sum is between 700 and 900
• 294 x 53
Lowest range: 200 x 50 = 10,000
Highest range: 300 x 60 = 18,000
The product is between 10,000 and 18,000
3.2 One Column
Example:
• 498 + 251
400 + 200 = 600
The sum is about 600
• 376 + 53 + 417
300 + 0 + 400 = 700
The sum is about 700
3.3 Two Column
Example:
• 458 + 251
450 + 250 = 700
The sum is about 700
• 376 + 53 + 417
370 + 50 + 410 = 830
The sum is about 830
3.4 With Adjustment
Example:
• 498 + 251
400 + 200 = 600
98 + 51 à 100 + 50 = 150
The sum is about 750
• 376 + 53 + 417
300 + 0 + 400 = 700
76 + 53 + 17 à 80 + 50 + 20 = 150
The sum is about 850
4. Rounding and Chopping
Example:
• 189.24 + 315.68 (To the nearest one decimal place)
Rounding Chopping
189.24 à 189.2 189.24 à 189.2
315.68 à 315.7 315.68 à 315.6
The sum is about 504.9 The sum is about 504.8
• + (To the nearest two decimal places)
Rounding Chopping
~ 453.20 ~ 453.20
~ 68.67 ~ 68.66
It is about 521.87 It is about 521.86
EXERCISE 3
1. Estimate the following by clustering:
a. 97.15 + 98.34 + 100.12 + 101.02
b. 200.17 + 198.23 + 199 + 202 + 209.11 + 197.99
c.
d.
2. Estimate the following by compatible number:
a. 124.56
b.
c.
d.
3. Estimate the following by Front-End Estimating (i) range, (ii) one column, (iii) two column, and (iv) with adjustment.
a. 254 + 34 - 312
b. 35 x 98
c. 3 467 + 5 456 + 8 712
d. 5668 x 321
4. Estimate the following by rounding and chopping:
a. 67.89 + 65.34 (two the nearest 1 place decimal)
b. 97.45 + 20.15 – 49.89 (to the nearest 1 place decimal)
c. 300.972 – 99.9832 (to the nearest 2 place decimals)
d. 0.9963 + 101.111 + 20.3415 (to the nearest 3 place decimals)
REFERENCES
David Phillips, Jenny Watson, Elena Iampolsky, Sonja Stambulic, Math Quest 7For Victoria: CSF Level 5, 2000, John Wiley & Sons Australia, Ltd. 33 Park Road, Milton, Qld 4064
Teh Keng Song and Looi Chin Keong, New Syllabus MATHEMATICS 1, 2006, Shinglee Publisher Pte Ltd. 120 Hillview Avenue, Kewalram Hillview Singapore 669594
Yusuf Fuad, Estimation or Approximation, a Presentation at SBI Workshop, 2007, Not Published, Yogyakarta
ESTIMATION
ESTIMATION
Based on the Oxford Advanced Learner’s Dictionary there are two definitions of estimation.
The first definition of estimation is a judgement or opinion about the value or quality of somebody or something. For example, who is the best candidate in your estimation?
Meanwhile, the second definition of estimation is a judgement about the levels or quantity of something. For example, the estimation of our total sales is around 10 million.
Now study the following explanation.
How would listeners react if a football commentator announced that there were 48 271 people sitting in the stands at the Gelora Bung Karno waiting for the football match to begin? Does anyone care?
It is more usual to hear that there are 48 000 or 50 000 spectators as it is often not necessary to know the exact number of people. An estimate is enough, so the nearest rounded number is used.
We often need to use estimation when getting a precise answer is impossible, unnecessary, or inconvenient.
Estimation is not the same as a guess because it is based on information.
For example, we may know how many people are able to fit into the football ground and the approximate percentage of seats filled. We can use this information to produce our estimate.
In order to get the best estimation, there are some estimation strategies as follows:
• Estimate by Rounding
• Estimate by Clustering
• Estimate by Using Compatible Numbers
• Estimate by Using Front-End Estimation
• Rounding and Chopping
In this chapter we will discuss deeply about rounding and estimate by rounding. Meanwhile the others strategies will explain in brief as enrichment.
ROUNDING
In rounding strategies, we may round up, round down or round off to the nearest. For example, we round up Rp 85,000.00 to Rp 100,000.00 when we budget for a trip that will cost at least Rp 85,000.00. At NTUC supermarket, Singapore, the bill is round down to the nearest 5 cents. For example, if our bill is $ 12.03 then we pay $ 12.00.
In mathematics we round off the number to the nearest.
· Include one extra digit for consideration.
· Simply drop the extra digit if it is less than 5.
· If it is 5 or more, add 1 to the previous digit before dropping the extra digit.
Example 1
Round off the following number to (i) the nearest whole number, (ii) 2 decimal places, and (iii) 3 decimal places.
a. 9.7168 b. 19.2147 c. 0.82514
Solution:
a. (i) 9.7168 10 (ii) 9.7169 9.72 (iii) 9.7168 9.717
This digit is more than 5
This digit is more than 5
This digit is more than 5
b. (i) 19.2147 19 (ii) 19.2147 19.21 (iii) 19.2147 19.215
This digit is more than 5
This digit is less than 5
This digit is less than 5
c.
This digit is less than 5(i) 0.82514 1 (ii) 0.82514 0.83 (iii) 0.82514 0.825
This digit is more than 5
This digit is 5
Example 2
Round off the following number to (i) the nearest hundred and (ii) the nearest ten.
a. 179 b. 139 c. 124
Solution:
a. (i) 179 200 (ii) 179 180
b. (i) 139 100 (ii) 139 140
c. (i) 124 100 (ii) 124 120
EXERCISE 1
1. Write the following correct to (1) the nearest whole number and (ii) 2 decimal places:
a. 5.42467 b. 15.824 c. 7.862 d. 130.629
2. Write the following correct to 3 decimal places:
a. 712.8926 b. 0.00272 c. 0.8274 d. 7.024489
3. Express as decimal and give your answer correct to 3 decimal places.
4. Express the following fractions as decimal correct to 2 decimal places.
a. b. c. d.
5. Round off the following to (i) the nearest ten and (ii) the nearest hundred:
a. 7 029 c. 5 624 c. 8 790 d. 956
ESTIMATE BY ROUNDING
Example 1
Estimate the result of
a. 189.2 315.6
b.
Solution:
a. 189.2 x 315.6
Round off each number to the nearest hundred.
189.2 à 200
315.6 à 300
Then multiply 200 x 300
So, the product of 189.2 x 315.6 is about 60,000
b.
Round off each number to the nearest ten.
à 450
à 70
Then add 450 + 70
So, the sum of is about 520
EXERCISE 2
1. Estimate the answers to each of these:
a. 5961 + 1768
b. 432 – 192
c. 48 022 538
d. 9701 x 37
e. 98 631 + 608 897
f. 6501 + 3790
g. 11 890 – 3642
h. 83 481 1751
i. 112 000 x 83
j. 66 501 738
k. 392 x 113 486
l. 12 476 24
2. During a sale, one kilogram of fish was sold for £ 4.95. Estimate how many kilograms of fish you could buy with £20.
3. Without doing an exact calculation, determine whether you can afford all the items below if you have only £30.
· 1 two-kilogram bottle of corn oil for £6.95.
· 5 cans of peach at £1.95 per can.
· 300 g of beef at £1.02 per 100 g.
· 24 pockets of recombined milk at £2.85 for 6.
4. Estimate the value of 52.97603 – 31.32186
5. Estimate the value of
a. b.
THE OTHER ESTIMATE STRATEGIES
1. Estimate by Clustering
Example:
• 99.7 + 97.83 + 102.18 + 100.101 + 99.98
All of the numbers are close to 100.
There are five numbers.
The sum is about 5 x 100 = 500
•
All of the numbers are close to 15.
There are four numbers.
The sum is about 4 x 15 = 60
2. Estimate by Using Compatible Numbers
Example:
• 76.36 24.73
76.36 is close to 75.
24.73 is close to 25.
The quotient of 76.36 24.73 is about 75 25 = 3
•
is close to 7
is close to 21
The sum of is about 40
3. Estimate by Using Front-End Estimation
There are 4 techniques of estimate by using Front-End Estimation.
3.1 Range
Example:
• 257 + 576
Lowest range: 200 + 500 = 700
Highest range: 300 + 600 = 900
The sum is between 700 and 900
• 294 x 53
Lowest range: 200 x 50 = 10,000
Highest range: 300 x 60 = 18,000
The product is between 10,000 and 18,000
3.2 One Column
Example:
• 498 + 251
400 + 200 = 600
The sum is about 600
• 376 + 53 + 417
300 + 0 + 400 = 700
The sum is about 700
3.3 Two Column
Example:
• 458 + 251
450 + 250 = 700
The sum is about 700
• 376 + 53 + 417
370 + 50 + 410 = 830
The sum is about 830
3.4 With Adjustment
Example:
• 498 + 251
400 + 200 = 600
98 + 51 à 100 + 50 = 150
The sum is about 750
• 376 + 53 + 417
300 + 0 + 400 = 700
76 + 53 + 17 à 80 + 50 + 20 = 150
The sum is about 850
4. Rounding and Chopping
Example:
• 189.24 + 315.68 (To the nearest one decimal place)
Rounding Chopping
189.24 à 189.2 189.24 à 189.2
315.68 à 315.7 315.68 à 315.6
The sum is about 504.9 The sum is about 504.8
• + (To the nearest two decimal places)
Rounding Chopping
~ 453.20 ~ 453.20
~ 68.67 ~ 68.66
It is about 521.87 It is about 521.86
EXERCISE 3
1. Estimate the following by clustering:
a. 97.15 + 98.34 + 100.12 + 101.02
b. 200.17 + 198.23 + 199 + 202 + 209.11 + 197.99
c.
d.
2. Estimate the following by compatible number:
a. 124.56
b.
c.
d.
3. Estimate the following by Front-End Estimating (i) range, (ii) one column, (iii) two column, and (iv) with adjustment.
a. 254 + 34 - 312
b. 35 x 98
c. 3 467 + 5 456 + 8 712
d. 5668 x 321
4. Estimate the following by rounding and chopping:
a. 67.89 + 65.34 (two the nearest 1 place decimal)
b. 97.45 + 20.15 – 49.89 (to the nearest 1 place decimal)
c. 300.972 – 99.9832 (to the nearest 2 place decimals)
d. 0.9963 + 101.111 + 20.3415 (to the nearest 3 place decimals)
REFERENCES
David Phillips, Jenny Watson, Elena Iampolsky, Sonja Stambulic, Math Quest 7For Victoria: CSF Level 5, 2000, John Wiley & Sons Australia, Ltd. 33 Park Road, Milton, Qld 4064
Teh Keng Song and Looi Chin Keong, New Syllabus MATHEMATICS 1, 2006, Shinglee Publisher Pte Ltd. 120 Hillview Avenue, Kewalram Hillview Singapore 669594
Yusuf Fuad, Estimation or Approximation, a Presentation at SBI Workshop, 2007, Not Published, Yogyakarta
