Topic 3: Approximations – Mathematics Form One
ABOUT THE TOPIC
Understanding approximations is important in our daily life. Approximations are used in various real life situations like estimating cost of items, distances, and ingredients for a recipe.
In this lesson, you will learn about rounding off numbers by place values, number of decimal places and by significant figures. Also, you will learn how to perform approximation in calculations.
The competencies developed will help you to perform real-life activities such as estimating number of things, analysing large data, making quick calculations, preparing budgets, and many other applications.
The Meaning of approximations
Approximation is a process of finding a number that is closest to the exact value. It involves expressing a number into a higher or lower value which is close to the exact value. Approximation can be termed as estimation and is represented by the symbol ‘≈’.
Activity 1: Estimating challenge
1. In your surrounding environment (class or home), estimate the number of objects or students available or measurements of objects.
2. Record your answers and perform actual counting or measurements and find the difference.
3. Compare the differences you have noted between the estimated and actual measurement and comment on the differences.
Rounding off numbers
Rounding off numbers is a process of making a number simpler, but keeping its value closer to the original value. The outcome is less precise but more user-friendly. Rounding off the number can be done by considering place values, number of decimal places, and number of significant figures.
Rounding off numbers by place values
Rounding off numbers by place values is done by considering the given place value for both whole and decimal parts of a number. Thus, for whole numbers, it can be done to the nearest ones, tens, hundreds, thousands or any other place value.
For decimal numbers, it can be done to the nearest tenths, hundredths, thousandths and so on. The following are basic procedures for rounding off numbers by place values.
(a) If the digit to the right of the rounding digit is either 0, 1, 2, 3, or 4, then the digit at the required place value remains unchanged, and each digit to the right of it become zeros.
Note: A rounding digit is the digit at a place where a number is supposed to be rounded off.
EXAMPLE 1
Round off each of the following numbers according to the given instruction.
(i) 591,622 to the nearest hundreds.
(ii) 35.4 to the nearest ones.
(iii) 234 to the nearest hundreds.
Solution
(i) The digit of 591,622 in the hundreds place is 6 and the digit on the right of 6 is 2, which is less than 5. In this case, maintain 6 and replace each digit to the right of 6 with zero. Thus, 591,622 to the nearest hundreds is 591,600.
(ii) 35.4 to the nearest ones is 35.
(iii) 234 to the nearest hundreds is 200.
EXAMPLE 2
Round off;
(i) 35.49643 to the nearest thousandths.
(ii) 274 to the nearest tens.
(iii) 3.2743 to the nearest hundredths.
(iv) 856,145 to the nearest thousands.
(v) 42.245468 to the nearest thousandths.
Solution
(i) 35.49643 rounded off to the nearest thousandths is 35. 496.
(ii) 274 rounded off to the nearest tens is 270.
(iii) 3.2743 rounded off to the nearest hundredths is 3.27.
(iv) 856,145 rounded off to the nearest thousands is 856,000.
(v) 42.245468 rounded off to the nearest thousandths is 42.245.
(b) If the digit to the right of a rounding digit is greater than 5, then 1 is added to the rounding digit and each digit to the right is replaced by zero.
EXAMPLE 3
Round off:
(i) 0.267 to nearest tenths.
(ii) 17.82 to the nearest ones.
(iii) 63 504 to the nearest thousands.
Solution
(i) 0.267 to the nearest tenths is 0.3.
(ii) 17.82 rounded off to the nearest ones is 18.
(iii) 63 504 rounded off to the nearest thousands is 64 000.
EXAMPLE 4
The population of a certain country in 1967 census showed that there were 5,834,875 men and 6,111,188 women. Round off the figures to the nearest millions.
Solution
5,834,875 men is rounded off to the nearest millions as 6,000,000 men (1 is added to 5 because the digit on the right of 5 is 8).
6,111,188 women is rounded off to the nearest millions as 6,000,000 women (6 remain unchanged since the digit on the right of 6 is 1 which is less than 5 and each digit on the right of 6 is replaced by zero).
(c) If the digit to the right of a rounding digit is 5, then the rounding digit is treated in three main ways.
(i) Add 1 to the rounding digit if it is odd.
(ii) Add 1 to the rounding digit (whether even or odd) if there are other digits that follow 5 which are all not zero.
(iii) The digit remains unchanged if all the digits following 5 are zeros and the rounding digit is even.
EXAMPLE 5
Round off each of the following numbers according to the given instructions.
(a) 2.635 to the nearest hundredths.
(b) 42,243,568 to the nearest thousands.
(c) 8,115,684 to the nearest ten thousands.
Solution
(a) 2.635 rounded off to the nearest hundredths is 2.64 since the rounding digit is odd and the digit to its immediate right is 5.
(b) 42,243,568 rounded off to the nearest thousands is 42,244,000 since digits after 5 (which is on the right of 3 as a rounding digit) are all not zero.
(c) 8,115,684 rounded to the nearest ten thousands is 8,120,000 since digits after 5 (which is on the right of 1 as a rounding digit) are all not zero.
EXAMPLE 6
Round off:.
(a) 6.645 to the nearest hundredths.
(b) 8,125,000 to the nearest ten thousands.
Solution
(a) 6.645 rounded off to the nearest hundredths is 6.64 since digits after 5 are all zeros.
(b) 8,125,000 rounded off to the nearest ten thousands is 8,120,000 since all the digits after 5 are all zero.
EXAMPLE 7
Round off the following numbers according to the instruction.
(a) 3.6265 to the nearest thousandths.
(b) 2,500,000 to the nearest millions.
Solution
(a) 3.6265 to the nearest thousandths is 3.626 since the digit to the right of the rounding digit (6) is 5 and all digits to the right of 5 are zeros.
(b) 2,500,000 to the nearest millions is 2,000,000 since the digit to the right of the rounding digit (2) is 5 and all digits to the right of 5 are zeros.
EXAERCISE 1
1. Round off each of the following numbers to the nearest thousands.
(a) 8,259 (f) 2,349,673
(b) 12,222 (g) 60,500
(c) 13,709 (h) 9,999
(d) 100,998 (i) 1,234,567
(e) 17,501
2. Round off each of the following numbers to the nearest ones.
(a) 41.4 (d) 0.8
(b) 0.49 (e) 0.379
(c) 2.613 (f) 2.55
3. Round off each of the following numbers to the nearest tens.
(a) 25.12
(b) 16.15
(c) 28.929
(d) 10.0089
(e) 50.408
(f) 33.456
(g) 20.17
(h) 311.114
4. The total mass of cotton harvested in a certain district was 17,816,273 kg. Round off the mass to the nearest.
(a) Millions of kilograms (b) Thousands of kilograms
5. In 1983, the number of primary school pupils in a certain region was 237,268. Round off the number of pupils to the nearest thousands.
6. Find the value of each of the following, and round off the answers to the nearest hundredths:
(a) 38.5 × 4.1
(b) 9.9 × 9.9
(c) 2.5 × 43.642
(d) 10.3 × 4.4
(e) 72 × 0.98
(f) 0.048 × 20.08
7. Find the value of each of the following, and round off the answers to the nearest tenths.
(a) 2,684 ÷ 17
(b) 4,314 ÷ 430
(c) 0.33 ÷ 1.1
(d) 5,301 ÷ 18
8. Convert each of the following fractions into decimals and round off the answers to the nearest tenths.
(a) 29 (b) 37 (c)173 (d) 14
9. Consider the following numbers: 9, 2.1, 42.045, and 635.7891.
(a) Add all the given numbers.
(b) Round off the number obtained in (a) to the nearest:
(i) Hundredths (ii)Tenths (iii) Hundreds
10. A factory got a profit of Tshs 67,459,853 after selling its products. How much profit did the factory get to the nearest thousands?
11. An electrician used 12,789 m, 13,952 m, and 9,374 m lengths of wire in 3 different months, respectively.
(a) Find the average amount of wire used to the nearest hundreds for the three months
(b) What is the difference between the actual amount and the estimated amount?
Rounding off by number of decimal places
Rounding off decimals involves a similar process of rounding numbers by place values. When rounding off decimals, the digits to the right of the given decimal place are dropped rather than being replaced by zeros.
In the process of rounding off, the following steps are used:
(a) If the digit after the rounding digit is either 0, 1, 2, 3, or 4, then the rounding digit remain unchanged, and all the other digits to the right are dropped.
EXAMPLE 8
Round off 0.2464 to 3 decimal places.
Solution
The digit in the third decimal place is 6 and the nearest digit to the right is 4 which is less than 5. In this case, 6 will remain unchanged and 4 is dropped.
Therefore, 0.2464 ≈ 0.246 to 3 decimal places.
(b) If the digit to the right of a rounding digit is greater than 5, then 1 is added to the rounding digit and other digits to the right are dropped.
EXAMPLE 9
Round off 0.97381 to 3 decimal places.
Solution
The digit in the third decimal place is 3 and the nearest digit to the right is 8 which is greater than 5. In this case, 1 is added to the rounding digit and each digit to the right is dropped. Therefore, 0.97381 ≈ 0.974 to 3 decimal places.
(c) If the digit to the right of a rounding digit is 5, then the rounding digit is considered as follows:
(i) Add 1 to the rounding digit if it is odd.
(ii) Add 1 to the rounding digit (whether even or odd) if there are other digits that follow 5 which are all not zero.
(iii) The digit remains unchanged if all the digits following 5 are zeros and the rounding digit is even.
EXAMPLE 10
Round off the following numbers.
(i) 6.2358 to two decimal places.
(ii) 3.4655 to two decimal places.
(iii) 0.2475 to three decimal places.
Solution
(i) 6.2358 rounded off to two decimal places is 6.24 since the rounding digit is odd and the digit to the right of the rounding digit is 5.
(ii) 3.4655 is rounded off as 3.47 to two decimal places since the rounding digit is followed by 5 and the digit after 5 are not all zero.
(iii) 0.2475 is rounded off as 2.248 to three decimal places since the rounding digit is odd.
EXAMPLE 11
Round off the following numbers.
(i) 1.88500 to two decimal places.
(ii) 2.682450 to four decimal places.
Solution
(i) 1.88500 rounded off to two decimal places is 1.88 since the digits after 5 are all zero.
(ii) 2.682450 rounded off to four decimal places is 2.6824 as the digit after 5 is 0.
EXAMPLE 12
i) Round off 8.6521 to one decimal place.
(ii) Round off 81.26584 to two decimal places.
Solution
(i) 8.6521 is rounded off as 8.7 to one decimal place.
(ii) 81.26584 is rounded off as 81.27 to two decimal places.
EXAMPLE 13
(i) Round 3.485 to two decimal places.
(ii) Round 0.65 to one decimal place.
(iii) Round 0.2465 to three decimal places.
Solution
(i) 3.485 is rounded off as 3.48 to two decimal places.
(ii) 0.65 is rounded off as 0.6 to one decimal place.
(iii) 0.2465 is rounded off as 0.246 to three decimal places.
EXERCISE 2
1. Round off the following numbers correct to the given number of decimal places.
(a) 4000.001 (2 decimal places)
(b) 19.86 (1 decimal place)
(c) 0.0075 (3 decimal places)
(d) 25.35447 (4 decimal places)
(e) 0.01001 (4 decimal places)
(f) 10.0100165 (3 decimal places)
2. Find the value of each of the following correct to 2 decimal places.
(a) 3.142 × 15.25 × 15
(b) 7.9909 × 0.011
(c) 0.0073 × 0.73
3. Write each of the following numbers correct to 2 decimal places.
(a) 0.0817
(b) 5.0744
(c) 1.70007
(d) 0.7153
(e) 12.047
(f) 3.3456
(g) 3.6149
(h) 0.00825
(i) 72.7946
4. Convert each of the following fractions into decimals correct to 3 decimal places.
(a) 107 (b) 49 (c) 23 (d) 411 (e) 117 (f) 21129
5. Write each of the following numbers correct to 1 decimal place.
(a) 3.142 (d) 0.09178
(b) 0.6667 (e) 0.7159
(c) 250.707 (f) 10.445
6. Write each of the following numbers correct to 3 decimal places.
(a) 0.7526
(b) 34.7007
(c) 3.14159
(d) 8.4999
(e) 5.5555
(f) 2.66732
7. Divide 8.51 by 0.472, then give the answer correct to 2 decimal places.
8. Multiply 9.0017 by 0.0987, then give the answer correct to 2 decimal places.
9. Round off the following numbers correct to the given number of decimal places.
(a) 0.002752 (4 decimal places)
(b) 20.04416 (2 decimal places)
(c) 0.17244 (3 decimal places)
(d) 6.0097 (3 decimal places)
(e) 2.14678 (1 decimal place)
10. Using a calculator or otherwise, evaluate the following and give the answer in specified number of decimal places.
(a) 27×3.6484.8954 (2 decimal places)
(b) 10.64×13.2814.32×4.893 (3 decimal places)
(c) 2.84+1.26.28−3.49 (1 decimal place)
(d) 153.4 ÷ 6.5 (1 decimal place)
(e) 44.0561 ÷ 11 (2 decimal places)
(f) 0.001144 ÷ 0.056 (3 decimal places)
11. If the length of a square plate is 3.14 m, calculate its area correct to 3 decimal places.
12. Evaluate each of the following expressions and write your answer according to the given instructions.
(a) 531.7 ÷ 4.3 (1 decimal place)
(b) 21.0561 ÷ 14 (1 decimal place)
(c) 0.001432 ÷ 0.076 (3 decimal places)
Significant figures
A significant figure is a digit in a number that gives information about the precision and accuracy of the measured value.
Rules for significant figures
1. Any digit from 1 to 9 appearing in a number is a significant figure.
2. Each zero appearing in a number between digits 1 to 9 is a significant figure. For example, 0 in 602 is a significant figure. That is, 602 has 3 significant figures.
3. When zeros are written to the right of the last non-zero digit, the zeros are not significant figures. For example, 72,000 has only 2 significant figures.
4. In decimals, any zero to the left of the first non-zero digit is not a significant figure. For example, the zeros in 0.025 are not significant figures. Thus, 0.025 has 2 significant figures.
5. All zeros that are on the right of a decimal point are significant only if a non-zero digit does not follow them. For example, 0.040 has two significant figures while 0.6700 has four significant figures, and 10.000 has five significant figures.
6. When zero is written at the end of an approximated number, including decimals, it is a significant figure. For example, 2.73 ≈ 3.0, and in this case, 0 is a significant figure. Also, 45,961 ≈ 46,000, the zero at the third-place value is a significant figure.
7. All the zeros that are on the right of the non-zero digit are significant if they come from measurements. For instance, 20 m has two significant figures and 300 g has 3 significant figures.
Note:
(i) Significant figures are also known as significant digits.
(ii) Significant figures are always counted from left towards the right of a number.
Rounding off numbers by significant figures
Rounding off numbers to a specified number of significant figures is one of the methods of approximations. Similar rules for rounding off numbers are used in writing numbers to a given number of significant figures.
EXAMPLE 14
Determine the number of significant figures in each of the following numbers.
(a) 2.3004 (d) 4,002,000,000
(b) 0.0804 (e) 0.00165000
(c) 0.00002 (f) 3,000 km
Solution
(a) 2.3004 has 5 significant figures.
(b) 0.0804 has 3 significant figures.
(c) 0.00002 has 1 significant figure.
(d) 4,002,000,000 has 4 significant figures.
(e) 0.00165000 has 6 significant figures.
(f) 3,000 km has 4 significant figures.
EXAMPLE 15
Write 38,176 correct to 1 significant figure.
Solution
Counting from left to right, the given number has 5 significant figures. The first digit, 3 is increased by one to become 4 (since 8 is greater than 5), and each of the right-side digits become zero. Thus, 38,176 ≈ 40,000 correct to 1 significant figure.
EXAMPLE 16
Find the value of 0.3143 × 6.06 giving the answer correct to 3 significant figures.
Solution
0.3143 × 6.06 = 1.904658
≈ 1.90
Therefore, 0.3143 × 6.06 ≈ 1.90 correct to 3 significant figures.
EXAMPLE 17
EXERCISE 3
1. Determine the number of significant figures from each of the following numbers.
(a) 5. 0372
(b) 97168.90000
(c) 0.0000678
(d) 292.00044
(e) 1.0000678
(f) 0.101
(g) 0.001110000
(h) 12030 km
2. Approximate each of the following numbers correct to the required number of significant figures.
(a) 0.285173 (3 significant figures)
(b) 88092.7 (4 significant figures)
(c) 2.007138 (3 significant figures)
(d) 74.471 (1 significant figure)
(e) 10.6987 (3 significant figures)
(f) 126. 306 (2 significant figures)
Approximations in calculations
When working out some calculations involving numbers, sometimes it is important to find an estimate of the answer. To find an estimated answer, take a suitable approximation by rounding off the numbers involved.
A suitable estimation can be to round off a number to the nearest whole number or in simple decimals which enables you to estimate the given problem easily and quickly.
EXAMPLE 18
Estimate the value of 38 × 71.
Solution
For easy estimation, round off 38 and 71 to the nearest tens.
That is, 38 × 71 ≈ 40 × 70 = 2 800 .
Therefore, 38 × 71 ≈ 2 800.
EXAMPLE 19
Estimate the value of 256.5 ÷ 63.5
Solution
Round off both numbers to the nearest ones.
256.5 ≈ 256 to the nearest ones and 63.5 ≈ 64 to the nearest ones. It follows that,
256.5 ÷ 63.5 ≈ 256 ÷ 64
≈ 4
Therefore, 256.5 ÷ 63.5 ≈ 4.
EXAMPLE 20
EXAMPLE 21
EXAMPLE 22
EXERCISE 4
In question 1 to 10, estimate the value of each of the given expressions.
1. 43 × 28
2. 2 912 × 32
3. 82 × 61
4. 868 × 31
5. 2.94 × 248
6. 171 220 ÷ 79
7. 35 164 × 23.04
8. 1.029 ÷ 0.021
9. 4 981 ÷ 6 438
10. 9 110 218 800 ÷ 4 081
11. A shopkeeper sold 192 t-shirts at a price of 5,950 Tanzanian shillings each. By estimation, how much money did the shopkeeper get?
12. A village received 40,376 bags of fertilizer to be distributed to 392 farmers. Estimate the number of bags each farmer got.
13. Estimate the answers of each of the following expressions.
(a) 25.45 ÷ 268
(b) 1.0031 + 56.241
(c) 528 × 3 902
(d) 61 ÷ 24
(e) 824 − 325
Chapter summary
1. Approximating or rounding off a number is a process of writing a number close enough to the exact number.
2. During approximation, if the digit to the right is 5, the rounding digit is:
(i) increased by 1 if it is odd.
(ii) left unchanged if it is even and has zero digits after 5 or has no digit after 5.
(iii) Any digit from 1 to 9 appearing in a number is a significant figure.
(iv) When zeros are written to the right of the last non-zero digit of an exact number, the zeros are not significant figures.
3. In decimals, any zero written to the left of the first non-zero digit is not a significant figure.
4. All zeros that are on the right of the decimal point are significant, only if a non-zero digit does not follow them.
5. When a zero is written at the end of an approximated decimal or number, it is a significant figure.
6. All the zeros that are on the right of the last non-zero digit are significant if they come from measurements.
Revision exercise
1. Round off each of the following numbers to the nearest hundredths.
(a) 8.648 (d) 31.7842
(b) 1.0544 (e) 19.6723
(c) 0.341 (f) 0.453
2. Round off each of the following numbers to the nearest millions and thousands.
(a) 78,911,393
(b) 1,114,562
(c) 22,878,130
(d) 1,350,095,450
(e) 20,781,233
3. The population of children under the age of fifteen years in a certain country in 2004 was 5,267,910. Round off the population to the nearest.
(a) Millions
(b) Thousands
(c) Hundreds
5. Write 86.463 correct to:
(a) 1 decimal place
(b) 2 decimal places
(c) 1 significant figure
(d) 2 significant figures
6. Write 0.00607049 correct to:
(a) 3 decimal places
(b) 4 decimal places
(c) 5 decimal places
(d) 6 decimal places
7. (a) Write each of the following numbers correct to 1 decimal place.
(i) 17.84 (ii) 17.084 (iii) 2.045 (iv) 0.048
(b) Write each of the following numbers correct to 2 decimal places.
(i) 23.748 (ii) 23.0845 (iii) 0.0485. (iv) 0.0803
8. Write the following numbers correct to 2 significant figures.
(a) 8 034 000
(b) 0.47
(c) 0.0785
(d) 0.82500
(e) 1.0271
(f) 6.0450
(g) 0.008
(h) 0.307
(i) 0.379
10. Determine the number of significant figures in each of the following numbers.
(a) 2.73
(b) 400 780
(c) 0.006
(d) 0.1089
(e) 0.12000
(f) 4 800 kg
11. Write:
(a) 34.996 correct to 2 decimal places.
(b) 35.0482 correct to 3 significant figures.
12. Write each of the following numbers correct to three decimal places.
(a) 0.00606
(b) 3.199281
(c) 8.27491
(d) 2.6047
(e) 72.87247
14. Find the value of each of the following expressions.
(a) 69 875 × 12 + 789 852 ÷ 12 (to the nearest thousands)
(b) 134 048 ÷ 568 − 96 045 + 279 455 ×18 (to the nearest hundred thousands)
(c) 155 × (98654168 − 92652149) (to the nearest millions)
(d) 2618954 + (45260 ÷ 365) × 68 − (1614312 − 1594069) (to the nearest thousands)
(e) 895 + (5 894 325 ÷ 85 425) − 794 (to the nearest tens)
15. (a) A certain boys’ school with 2,485 students has decided to make uniforms for all students. If a shirt and a pair of trousers use 2 m and 1.5 m, respectively, how much materials of cloth in total will the school need? Give your answer to the nearest hundreds.
(b) Two hundred and forty-six university students were given Tshs 21,069,900 to share equally among themselves. How much money did each student get to the nearest thousands?
(c) A train can carry 150 passengers in one trip. If it makes 16 trips a day
(i) Estimate to the nearest ten thousands the number of passengers the train can carry in 5 days.
(ii) How much money to the nearest millions will be collected for 16 days when each passenger pays a fare of Tshs 1,050?
18. Write 47628.02415 correct to:
(a) 3 significant figures
(b) 2 decimal places
(c) 3 decimal places
(d) 2 significant figures
(e) 4 decimal places
(f) 4 significant figures
19. There are 64 pipes each with a length 4.22 m arranged horizontally end to end. Estimate the length of all the pipes.
20. A car uses 1.13 litres of petrol to travel a distance of 11.65 km.
(a) Estimate the rate of fuel consumption of the car in km per litre.
(b) Estimate the amount of fuel the car will consume for travelling:
(i) 40.28 km
(ii) 348 km
21. The XYZ cooperative society produces 94 jars of liquid soap every day. The cost of production of each jar consists of Tshs 4,015 for labour and Tshs 2,856 for materials. Estimate:
(a) The total cost of producing a jar of liquid soap.
(b) The cost of materials for producing 94 jars.
(c) The cost of labour for producing 94 jars.
22. Estimate the values obtained from each of the given expressions.
(a) 8.7 × 410
(b) 4.17 × 730
(c) 28 × 0.83
(d) 54. 5 × 1.96
(e) 430 ÷ 31.2
(f) 24.4 ÷ 0.673
(g) 9.05 ÷ 18.2
(h) 0.633 × 425
(i) 0.136 ÷ 8.45
(j) 7.40 × 36.4
Project: Estimating the value of pi.
1. Collect different cylindrical objects such as bottle tops, buckets, rods, pipes, and related objects.
2. Use a piece of thread to measure and carefully record the circumference and the diameter of at least 5 different cylindrical shapes.
3. Divide the circumference by the diameter obtained in task 2 and record your answers to 5 decimal places.
4. Calculate the average value of the results obtained in task 3.
5. Estimate the value obtained in task 4 correct to 3 decimal places.
6. The resulting number you obtained in task 5 is the approximated value of π which in fraction is 22/7
7. Explore various sources such as books and the internet to confirm the value of π.
8. Comment on your accuracy in estimating the value of π and discuss any possible ways you could reduce errors as much as possible.
