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# Chapter 4: Approximation

Last year you learnt that an approximate answer is almost correct, but not exact. Approximation is the process of using rounding to determine a fairly accurate value. Approximation is very useful in situations where you have to do a calculation quickly. Approximation is also used in situations where it does not make sense to work with the exact value, for example when a value has many decimal digits.

approximation Approximation is the process of using rounding to determine a fairly accurate value.

## 4.1 Rounding off

Rounding off is the basis of approximation. When we round off numbers, we make them less exact, but easier to work with. We usually round off a number to a specific digit or multiple of ten.

From previous years you know that when we are rounding off (in the base ten number system):

• We round down for the digits 0, 1, 2, 3, 4, which means the previous place value digit stays the same.
• We round up for the digits 5, 6, 7, 8, 9, which means the previous place value digit increases by 1.

rounding off Rounding off is the process of making a number less exact, but easier to work with, by adjusting it to a specific digit or multiple of ten.

round down When we are rounding off, we round down for the digits 0, 1, 2, 3, 4, by keeping the previous place value digit the same.

round up When we are rounding off, we round up for the digits 5, 6, 7, 8, 9, by increasing the previous place value digit by 1.

### Rounding off to a given multiple of ten

When we round off a number to a specific multiple of ten, we have to identify the digit that represents that multiple, and then determine whether the number must be rounded down or rounded up. You did this last year.

### Worked example 4.1: Rounding off to a given multiple of ten (Method 1)

A handbag is advertised for ₦2,951. Round off this number to the nearest hundred.

1. Step 1: Write down half of the required multiple of ten.

2. Step 2: Determine which digit represents the required multiple of ten.

You may use a place value table to help you with this:

\begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline \text{billion} & \text{hm} & \text{tm} & \text{million} &\text{hth} & \text{tth} & \text{thousand} & \text{ h } & \text{ t } & \text{ u } \newline \hline & & & & & & 2 & 9 & 5 & 1 \newline \hline \end{array}

The 9 represents the hundreds.

3. Step 3: Consider the digits to the right of the one you identified in Step 2. If they are less than the value in Step 1, the digit you identified remains the same. If they are equal to or more than the value in Step 1, the digit you identified increases by 1.

The digits to the right of the 9 are 51.

The 9 must increase by 1.

4. Step 4: If the digit that must increase is a 9, it changes to a 0. The digit to its left increases by 1.

The 9 must change to a 0.

The 2 to the left of the 9 becomes $2+1=3$.

5. Step 5: Write down the rounded number with the digits you determined in Step 4, with all the digits to the right of them as zeros.

₦3,000

If we round off 2,951 to the nearest hundred, we actually want to know whether this number is closer to 2,900 or to 3,000. The number 2,951 is just over halfway between 2,900 and 3,000. We therefore round up to the higher number.

### Worked example 4.2: Rounding off to a given multiple of ten (Method 2)

On a specific day, the population of Nigeria is 201,195,431. Round off this number to the nearest thousand.

1. Step 1: Circle the digit that represents the required multiple of 10.

Remember, from right to left the place values are:

unit, ten, hundred, thousand
ten thousand, hundred thousand, million
ten million, hundred million, billion
and so on â€¦

2. Step 2: Look at the digit to the right of the one you circled. If it is 0, 1, 2, 3 or 4, the circled digit remains the same. If it is 5, 6, 7, 8 or 9, the circled digit increases by 1.

The 5 remains the same.

3. Step 3: Write down the original number with the digit you determined in Step 2, and all the digits to the right of it as zeros.

201,195,000

If we round off 201,195,431 to the nearest thousand, we actually want to know whether this number is closer to 201,195,000 or 201,196,000. The number 201,195,431 is 431 units away from 201,195,000 and 569 units away from 201,196,000. So it is closer to 201,195,000.

### Exercise 4.1: Round off to a given multiple of ten

Round off each of the following numbers to the required multiple of 10.

1. 26,548 to the nearest ten

26,550

2. 48,849 to the nearest hundred

48,800

3. 37,627 to the nearest thousand

38,000

4. 58,503 to the nearest thousand

59,000

5. 29,618 to the nearest thousand

30,000

6. 5,319,432 to the nearest thousand

5,319,000

7. 19,553 to the nearest thousand

20,000

8. 2,905,047 to the nearest hundred

2,905,000

9. 5,095 to the nearest ten

5,100

10. 512,909,050 to the nearest hundred

512,909,100

### Rounding off to a given unit of measurement

If you are an engineer building a national highway, you are not going to measure the length of the road in millimetres! You are probably going to measure to the nearest hundred meters. Similarly, if you are a shop owner who sells rice by the cup, you cannot measure to the nearest kilogram. You have to measure accurately to the nearest gram.

Last year you used approximation for the following quantities. You also learned what each quantity's SI unit is:

• distances and dimensions, measured in metres (m)
• capacity, measured in litres (L)
• mass, measured in kilograms (kg).

You also worked with money: ₦1 = 100 kobo

SI unit An SI unit is a basic unit of measurement that forms part of the International System of Units.

### Worked example 4.3: Rounding off to a given unit of measurement

A recipe to bake bread gives the mass of flour needed as 482.52 g. Your kitchen scales can only measure accurately to the nearest gram. Determine the mass of flour you will measure out.

1. Step 1: Circle the first digit to the left of the decimal point.

2. Step 2: Look at the digit to the right of the decimal point. If it is 0, 1, 2, 3 or 4, the circled digit remains the same. If it is 5, 6, 7, 8 or 9, the circled digit increases by 1.

The 2 becomes $2+1=3$.

3. Step 3: Write down the original number with the digit you determined in Step 2, but without the decimal digits. Include the unit of measurement.

483 g

If your kitchen scales can only measure to the nearest gram, you actually want to know whether 482.52 g is closer to 482 g or 483 g. The fraction of a gram, 0.52 g, is just over halfway between 482 g and 483 g. That is why we round off to 483 g.

We can also round off a measurement to a multiple of ten or to part of a unit. For example, we can round 482.52 g to:

• the nearest ten grams: 480 g
• the nearest tenth of a gram: 482.6 g

### Worked example 4.4: Rounding off to a given fraction of a unit of measurement

A student must make a scale drawing. She calculates that the length of one of the lines must be 15.362 cm. Most rulers can only measure accurately to the nearest millimetre, which is one tenth of a centimetre. Determine the length of the line she will measure on her ruler.

1. Step 1: Circle the digit that represents the fraction of the unit of measurement.

You may use a place value table to help you with this:

\begin{array}{|c|c|c|c|c|c|} \hline \text{tens} & \text{units} & \text{.} & \text{ tenths } & \text{ hundredths } & \text{ thousandths } \newline \hline 1 & 5 & . & 3 & 6 & 2 \newline \hline \end{array}

Remember, from left to right after the decimal point the place values are:
tenth, hundredth, thousandth, and so on â€¦

The first digit after the decimal point is the one that must be rounded, because it represents one tenth of a centimetre (which is a millimetre).

2. Step 2: Look at the digit to the right of the one you circled. If it is 0, 1, 2, 3 or 4, the circled digit remains the same. If it is 5, 6, 7, 8 or 9, the circled digit increases by 1.

The 3 becomes $3+1= 4$.

3. Step 3: Write down the original number with the digit you determined in Step 2, but without the rest of the decimal digits. Include the unit of measurement.

15.4 cm

The student actually wants to know whether she must measure 15.3 cm or 15.4 cm on her ruler. The fraction of a centimetre, 0.362 cm, is over halfway between 0.3 cm and 0.4 cm. That is why we round off to 15.4 cm.

### Exercise 4.2: Round off to a given degree of accuracy

Round off each of the following measurements to the given degree of accuracy.

1. 25.329 g to the nearest gram

25 g

2. 308.721 ml to the nearest millilitre

309 ml

3. 75.5 cm to the nearest centimetre

76 cm

4. 348.531 ml to the nearest tenth of a millilitre

348.5 ml

5. 103.279 g to the nearest tenth of a gram

103.3 g

6. 115.555 cm to the nearest tenth of a centimetre

115.6 cm

7. 1.2109 g to the nearest hundredth of a gram

1.21 g

8. ₦519.85 to the nearest naira

₦520

9. 1,218.25 ml to the nearest 10 ml

1,220 ml

10. ₦439.99 to the nearest ten naira

₦440

### Rounding off to a required number of decimal places

If you press $\sqrt 2$ on your calculator, it displays 1.414213562. If your calculator window could display more decimal places, it would give even more digits after the decimal point. This is an example of a non-rational or irrational number. It has an unending number of digits after the decimal point that do not form any pattern. It is not practical to write down or work with so many digits after a decimal point. That is why you need to know how to round numbers to a required number of decimal places.

### Worked example 4.5: Rounding off to a required number of decimal places

Round 0.059687 to three decimal places.

1. Step 1: Count the number of required decimal places after the decimal point. Circle that digit.

The 9 is three places after the decimal point.

2. Step 2: Look at the digit to the right of the one you circled. If it is 0, 1, 2, 3 or 4, the circled digit remains the same. If it is 5, 6, 7, 8 or 9, the circled digit increases by 1.

The 9 must increase by 1.

3. Step 3: If the digit that must increase is a 9, it changes to a 0. The digit to its left increases by 1.

The 9 must change to a 0.

The 5 to the left of the 9 becomes $5+1=6$.

4. Step 4: Write down the original number with the digit you determined in Step 3. You were asked for three decimal places, so you must write zeros after your rounded number so that you have three decimal places.

We have to round to three decimal places, so the rounded number must have three digits after the decimal point.

0.060

If you were asked to round this number to two decimal places, you would omit the zero and give the answer as 0.06.

### Exercise 4.3: Round off to a required number of decimal places

Round off each of the following numbers to the required number of decimal places.

1. 45.326 to one decimal place

45.3

2. 26.553 to one decimal place

26.6

3. 72.901 to one decimal place

72.9

4. 39.0293 to two decimal places

39.03

5. 53.788 to two decimal places

53.79

6. 0.05732 to two decimal places

0.06

7. 0.85916 to three decimal places

0.859

8. 3.01152 to three decimal places

3.012

9. 0.00991 to three decimal places

0.010

10. 9.995 to two decimal places

10.00

## 4.2 Significant figures

Significant figures are especially important when you are working with measurements. The significant figures in a number tell us how accurate a measurement is.

Suppose you see a measurement of 0.00205 km. This does not necessarily mean that someone measured a distance correctly to five decimal places. The person could have measured the distance in metres or centimetres and just converted is to kilometres: 0.00205 km = 2.05 m = 205 cm. Note that the digits 205 form part of each measurement, no matter which unit of measurement we convert to. This means that the digits 205 are the significant figures.

Suppose the population of Lagos State is 19,473,026. It does not really make sense to report such a large population to the last unit, because it changes fairly quickly. Depending on what the number will be used for, it is usually rounded to only a few significant figures. Typically a population of that size is rounded to three significant figures. This means we round it to the nearest hundred thousand, which gives 19,500,000 or 19.5 million. When we use this rounded number, it is not necessary to keep track of changes over short periods of time.

significant figures The significant figures in a number are the digits that give us information about how accurate the number is. We read them from the first non-zero digit to the right.

### Rounding off to a required number of significant figures

We work with the following basic rules when we round numbers to significant figures:

• Non-zero digits and zeros in between non-zero digits are significant figures.
• Leading zeros, that is, zeros to the left of the first non-zero digit, are not significant.
• Trailing zeros, that is, zeros after the last non-zero digit, are only significant when they appear after a decimal point.

### Worked example 4.6: Rounding off whole numbers to a required number of significant figures

Round 401,503 to three significant figures.

1. Step 1: Identify all the significant figures in the number. Circle the first significant figure.

Non-zero digits and zeros in between non-zero digits are significant figures.

This means all the digits are significant figures. The 4 is the first significant figure: $\enclose{circle}[mathcolor="green"]{4}01,503$

2. Step 2: Count from the number you circled in Step 1 to the right. Circle the digit at the place that represents the given number of significant figures.

The 1 is the third digit to the right of the first significant number. It is the third significant figure.

\begin{array}{c c c } \enclose{circle}[mathcolor="green"]{4} & 0 & \enclose{circle}[mathcolor="red"]{1}, & 5\; \; 0 \;\; 3 \newline ^1 & ^2 & ^3 \end{array}
3. Step 3: Follow normal rounding rules to determine whether the number your circled in Step 2 must remain the same or increase by 1.

The 1 becomes $1+1=2$.

4. Step 4: Write down the original number with the digit you determined in Step 3, but with all the digits to the right of it as zeros.

402,000

### Worked example 4.7: Rounding off decimal fractions to a required number of significant figures

Round 0.0061090 to three significant figures.

1. Step 1: Identify all the significant figures in the number.

• Non-zero digits and zeros in between non-zero digits are significant figures.
• Zeros to the left of the first non-zero digit are not significant.
• Zeros after the last non-zero digit are only significant when they appear after a decimal point.

This means the number has five significant figures. The 6 is the first significant figure: $0.00\enclose{circle}[mathcolor="green"]{6}1090$

2. Step 2: Count from the number you circled in Step 1 to the right. Circle the digit at the place that represents the given number of significant figures.

The first 0 is the third digit to the right of the first significant number.

\begin{array}{c c c c c} 0\;\;.\;\;0\;\; 0 &\enclose{circle}[mathcolor="green"]{6} & 1 & \enclose{circle}[mathcolor="red"]{0} & 9\;\;0 \newline &^1 & ^2 & ^3 & \end{array}
3. Step 3: Follow normal rounding rules to determine whether the number you circled in Step 2 must remain the same or increase by 1.

The 0 becomes $0+1=1$.

4. Step 4: Write down the original number with the digit you determined in Step 3, but with all the digits to the right of it as zeros. Only write down zeros if they are needed to get to the required number of significant figures.

The original number becomes 0.0061100.

We need three significant figures, so we do not write down the two zeros at the end.

### Exercise 4.4: Round off to a required number of significant figures

Round off each of the following numbers to the required number of significant figures.

1. 56,031 to three significant figures

2. 501,007 to three significant figures

3. 19,532 to two significant figures

4. 0.03127 to three significant figures

5. 0.00583 to two significant figures

6. 0.01007 to three significant figures

7. 2.0612 to two significant figures

8. 15.009 to four significant figures

9. 1.60967 to four significant figures

10. 0.00106080 to four significant figures

## 4.3 Practical applications

Approximation is an important skill. You will also use it in subjects other than Mathematics.

To indicate that a number you are giving is a rounded number, you may use the symbol $\approx$, which means "approximately equal to".

### Exercise 4.5: Apply approximation in practice

1. The ratio of the circumference of a circle to its diameter is $\pi$. If you press $\pi$ on your calculator, it displays 3.141592654. In a Mathematics assignment, you are told that you must work correctly to three decimal places. Determine what value you will use for $\pi$.

2. A nurse must prepare an injection for a patient. He calculates that he needs to measure out 5.592 ml, but his syringe can only measure to the nearest tenth of a millilitre. Determine what volume the nurse must measure with his syringe.

3. The population of a town is given is 149,805 on a specific day. The authorities need a figure that they can use for a few months. Round the population to the nearest ten thousand.

4. A tailor needs 2.59 m of fabric to make a dress. She wants to buy enough fabric for 15 dresses. If the fabric is sold by the metre, how many metres must she buy?

\begin{align} 2.59\text{ m}\times 15 &= 38.85\text{ m} \\ &\approx 39\text{ m} \end{align}
5. A certain space on the schoolgrounds has to be fenced. The fencing wire is sold in rolls of 10 m. The principal wants to know approximately how many rolls she needs to buy. She sends the Mathematics class to measure the distance that must be fenced. They measure 857.5 m. Work out how many rolls the principal must buy.

6. A manufacturer of powdered milk includes a disclaimer on the back of the tin that reads as follows: "The contents of this tin has a mass of 380 g, rounded to the nearest 10 g. Determine the maximum mass of milk powder that can be in the tin.

If the mass was rounded to the nearest 1 g, the maximum mass is 384 g.

(For 385 g and more, the mass would have been rounded to 390 g.)

7. At a factory that produces paper, the mass of reams of paper are determined before they leave the factory. There are supposed to be 250 sheets of paper, each with a mass of 4 g, in each ream. The scales indicate that a specific ream has a mass of 1,004.15 g. Determine whether the number of sheets of paper in the ream is correct or not.

There is one sheet of paper extra in the ream.

8. Chinyere, Ngozi and Oluchi decide that they are going to buy three handbags together and then take turns to use the handbags. The bags they decide on cost ₦3,369, ₦4,579 and ₦4,299. They want to know more or less how much each of them must contribute. Calculate the average cost of the bags to the nearest ten naira.

\begin{array}{r r} &_{1} \phantom{0} \; _{2} \; _{2}\, \phantom{0}\newline & ₦\,3\, ,\, 3\, 6\, 9 \newline + & ₦\,4\, ,\, 5\, 7\, 9 \newline + & ₦\,4\, ,\, 2\, 9\, 9 \newline \hline & ₦\,1 \, 2\, , \, 2\, 4\, 7\newline \hline \end{array}

\begin{align} \frac{ ₦12,247}{3}=&\, ₦4,082.\dot{3}\\ \approx &\, ₦4,080 \end{align}
9. In a Chemistry problem, you calculate that you need a mass of 0.002507 g of sodium chloride to make up a solution.

1. Round the mass to three decimal places.
1. Round the mass to three significant figures.
1. Explain why you cannot measure out the sodium chloride on a scale that can only measure accurately to the nearest gram.

If you round off the mass to the nearest gram, you get 0.

10. A biologist determines under the microscope that the width of a human hair is 0.0001095 mm. Round this number to three significant figures.

## 4.4 Summary

• Approximation is the process of using rounding to determine a fairly accurate value.
• Rounding off is the process of making a number less exact, but easier to work with, by adjusting it to a specific digit or multiple of ten:
• We round down for the digits 0, 1, 2, 3, 4, by keeping the next digit to the left the same.
• We round up for the digits 5, 6, 7, 8, 9, by increasing the next digit to the left by 1.
• It is not practical to write down or work with many digits after a decimal point, so we round off to a given number of decimal places.
• The significant figures in a number are the digits that give us information about how accurate the number is. We read them from the first non-zero digit to the right.
• We work with the following basic rules when we round numbers to significant figures:
• Non-zero digits and zeros in between non-zero digits are significant figures.
• Leading zeros, that is, zeros to the left of the first non-zero digit, are not significant.
• Trailing zeros, that is, zeros after the last non-zero digit, are only significant when they appear after a decimal point.
• To indicate that a number you are giving is a rounded number, you may use the symbol $\approx$.