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# Rounding Off

## 1.4 Rounding off (EMA8)

Rounding off a decimal number to a given number of decimal places is the quickest way to approximate a number. For example, if you wanted to round off $$\text{2,6525272}$$ to three decimal places, you would:

• count three places after the decimal and place a $$|$$ between the third and fourth numbers;

• round up the third digit if the fourth digit is greater than or equal to $$\text{5}$$;

• leave the third digit unchanged if the fourth digit is less than $$\text{5}$$;

• if the third digit is $$\text{9}$$ and needs to be rounded up, then the $$\text{9}$$ becomes a $$\text{0}$$ and the second digit is rounded up.

So, since the first digit after the $$|$$ is a $$\text{5,}$$ we must round up the digit in the third decimal place to a $$\text{3}$$ and the final answer of $$\text{2,6525272}$$ rounded to three decimal places is $$\text{2,653}$$.

The following video explains how to round off.

Video: 2DD8

## Worked example 4: Rounding off

Round off the following numbers to the indicated number of decimal places:

1. $$\dfrac{120}{99}=\text{1,}\dot{1}\dot{2}$$ to $$\text{3}$$ decimal places.

2. $$\pi =\text{3,141592653...}$$ to $$\text{4}$$ decimal places.

3. $$\sqrt{3}=\text{1,7320508...}$$ to $$\text{4}$$ decimal places.

4. $$\text{2,78974526}$$ to $$\text{3}$$ decimal places.

### Mark off the required number of decimal places

If the number is not a decimal you first need to write the number as a decimal.

1. $$\dfrac{120}{99} = \text{1,212}|121212\ldots$$

2. $$\pi =\text{3,1415}|92653\ldots$$

3. $$\sqrt{3}=\text{1,7320}|508\ldots$$

4. $$\text{2,789}|74526$$

### Check the next digit to see if you must round up or round down

1. The last digit of $$\frac{120}{99}=\text{1,212}|121212\dot{1}\dot{2}$$ must be rounded down.

2. The last digit of $$\pi =\text{3,1415}|92653\ldots$$ must be rounded up.

3. The last digit of $$\sqrt{3}=\text{1,7320}|508\ldots$$ must be rounded up.

4. The last digit of $$\text{2,789}|74526$$ must be rounded up.

Since this is a $$\text{9}$$ we replace it with a $$\text{0}$$ and round up the second last digit.

1. $$\dfrac{120}{99}=\text{1,212}$$ rounded to $$\text{3}$$ decimal places.

2. $$\pi =\text{3,1416}$$ rounded to $$\text{4}$$ decimal places.

3. $$\sqrt{3}=\text{1,7321}$$ rounded to $$\text{4}$$ decimal places.

4. $$\text{2,790}$$

Exercise 1.2

Round off the following to $$\text{3}$$ decimal places:

$$\text{12,56637061...}$$

Mark off the required number of decimal places: $$\text{12,566}|37061\ldots$$. The next digit is a $$\text{3}$$ and so we round down: $$\text{12,566}$$.

$$\text{3,31662479...}$$

Mark off the required number of decimal places: $$\text{3,316}|62479\ldots$$. The next digit is a $$\text{6}$$ and so we round up: $$\text{3,317}$$.

$$\text{0,2666666...}$$

Mark off the required number of decimal places: $$\text{0,266}|6666\ldots$$. The next digit is a $$\text{6}$$ and so we round up: $$\text{0,267}$$.

$$\text{1,912931183...}$$

Mark off the required number of decimal places: $$\text{1,912}|931183\ldots$$. The next digit is a $$\text{9}$$ and so we round up: $$\text{1,913}$$.

$$\text{6,32455532...}$$

Mark off the required number of decimal places: $$\text{6,324}|55532\ldots$$. The next digit is a $$\text{5}$$ and so we round up: $$\text{6,325}$$.

$$\text{0,05555555...}$$

Mark off the required number of decimal places: $$\text{0,055}|55555\ldots$$. The next digit is a $$\text{5}$$ and so we round up: $$\text{0,056}$$.

Round off each of the following to the indicated number of decimal places:

$$\text{345,04399906}$$ to $$\text{4}$$ decimal places.

$\text{345,04399906} \approx \text{345,0440}$

$$\text{1 361,72980445}$$ to $$\text{2}$$ decimal places.

$\text{1 361,72980445} \approx \text{1 361,73}$

$$\text{728,00905239}$$ to $$\text{6}$$ decimal places.

$\text{728,00905239} \approx \text{728,009052}$

$$\dfrac{1}{27}$$ to $$\text{4}$$ decimal places.

We first write the fraction as a decimal and then we can round off.

\begin{align*} \frac{1}{27} &= \text{0,037037...} \\ & \approx \text{0,0370} \end{align*}

$$\dfrac{45}{99}$$ to $$\text{5}$$ decimal places.

We first write the fraction as a decimal and then we can round off.

\begin{align*} \frac{45}{99} &= \text{0,45454545...} \\ & \approx \text{0,45455} \end{align*}

$$\dfrac{1}{12}$$ to $$\text{2}$$ decimal places.

We first write the fraction as a decimal and then we can round off.

\begin{align*} \frac{1}{12} &= \text{0,08333...} \\ & \approx \text{0,08} \end{align*}

Study the diagram below

Calculate the area of $$ABDE$$ to $$\text{2}$$ decimal places.

$$ABDE$$ is a square and so the area is just the length squared.

\begin{align*} A &= l^{2} \\ &= \pi^2 \\ & = \text{9,86904...} \\ & \approx \text{9,87} \end{align*}

Calculate the area of $$BCD$$ to $$\text{2}$$ decimal places.

$$BCD$$ is a right-angled triangle and so we have the perpendicular height. The area is:

\begin{align*} A & = \frac{1}{2} b h\\ & = \frac{1}{2} \pi^2 \\ & = \text{4,934802...} \\ & \approx \text{4,93} \end{align*}

Using you answers in (a) and (b) calculate the area of $$ABCDE$$.

The area of $$ABCDE$$ is the sum of the areas of $$ABDE$$ and $$BCD$$.

\begin{align*} A & = \text{9,87} + \text{4,93} \\ & \approx \text{14,80} \end{align*}

Without rounding off, what is the area of $$ABCDE$$?

\begin{align*} A_{ABCDE} & = A_{ABDE} + A_{BCD} \\ & = l^{2} + \frac{1}{2}bh \\ &= \pi ^2 + \frac{1}{2} \pi^2 \\ &= \text{14,8044...} \end{align*}

Given $$i = \dfrac{r}{600}$$; $$r = \text{7,4}$$; $$n = 96$$; $$P = \text{200 000}$$.

Calculate $$i$$ correct to $$\text{2}$$ decimal places.

\begin{align*} i & = \frac{r}{600} \\ & = \frac{\text{7,4}}{600} \\ & = \text{0,01233} \\ & \approx \text{0,01} \end{align*}

Using you answer from (a), calculate $$A$$ in $$A = P(1+ i)^n$$.

\begin{align*} A &= P(1+ i)^n \\ &= \text{200 000}\left(1+ \text{0,01}\right)^{96} \\ &= \text{519 854,59} \end{align*}

Calculate $$A$$ without rounding off your answer in (a), compare this answer with your answer in (b).

\begin{align*} A &= P(1+ i)^n \\ A &= \text{200 000}\left(1+ \frac{\text{7,4}}{600}\right)^{96} \\ &= \text{648 768,22} \end{align*}

There is a $$\text{128 913,63}$$ difference between the answer in (b) and the one calculated without rounding until the final step.

If it takes $$\text{1}$$ person to carry $$\text{3}$$ boxes, how many people are needed to carry $$\text{31}$$ boxes?

Each person can carry 3 boxes. So we need to divide 31 by 3 to find out how many people are needed to carry 31 boxes.

$\frac{31}{3} = \text{10,3333...}$

Therefore $$\text{11}$$ people are needed to carry $$\text{31}$$ boxes. We cannot have $$\text{0,333}$$ of a person so we round up to the nearest whole number.

If $$\text{7}$$ tickets cost $$\text{R}\,\text{35,20}$$, how much does one ticket cost?

Since 7 tickets cost $$\text{R}\,\text{35,20}$$, 1 ticket must cost $$\text{R}\,\text{35,20}$$ divided by 7.

$\frac{\text{35,20}}{7} =\text{5,028571429}\\$

Therefore one ticket costs $$\text{R}\,\text{5,03}$$. Money should be rounded off to $$\text{2}$$ decimal places.