Solutions to Exercise 3A
Approximation and Estimation
Basic Level
1. Round off $698\,352$ to the nearest:
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(a) $100$
To round to the nearest 100, we look at the tens digit, which is 5. Since $5 \geq 5$, we round the hundreds digit (3) up. The number becomes $698\,400$.
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(b) $1000$
To round to the nearest 1000, we look at the hundreds digit, which is 3. Since $3 < 5$, we keep the thousands digit (8) as it is. The number becomes $698\,000$.
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(c) $10\,000$
To round to the nearest 10\,000, we look at the thousands digit, which is 8. Since $8 \geq 5$, we round the ten thousands digit (9) up. This causes a ripple effect, rounding the hundred thousands digit (6) up as well. The number becomes $700\,000$.
2. Correct $45.7395$ to:
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(a) 1 decimal place
The first decimal place is 7. We look at the next digit, which is 3. Since $3 < 5$, we keep 7 as it is. The number becomes $45.7$.
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(b) the nearest whole number
The whole number is 45. We look at the first decimal place, which is 7. Since $7 \geq 5$, we round up. The number becomes $46$.
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(c) 3 decimal places
The third decimal place is 9. We look at the next digit, which is 5. Since $5 \geq 5$, we round up. The number becomes $45.740$.
Intermediate Level
3. The dimensions of a rectangular plot of land are $28.3 \text{ m}$ by $53.7 \text{ m}$. Find:
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(i) the perimeter of the land, correct to the nearest $10 \text{ m}$.
Perimeter $P = 2(l + w) = 2(28.3 + 53.7) = 2(82) = 164 \text{ m}$.
To the nearest 10 m, we look at the units digit, which is 4. Since $4 < 5$, we round down. The perimeter is $160 \text{ m}$.
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(ii) the area of grass needed to fill up the entire plot of land, correct to the nearest $100 \text{ m}^2$.
Area $A = l \times w = 28.3 \times 53.7 = 1519.71 \text{ m}^2$.
To the nearest 100 m$^2$, we look at the tens digit, which is 1. Since $1 < 5$, we round down. The area is $1500 \text{ m}^2$.
4. Round off:
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(a) $4.918 \text{ m}$ to the nearest $0.1 \text{ m}$
The first decimal place is 9. The next digit is 1. Since $1 < 5$, we keep 9. The number becomes $4.9 \text{ m}$.
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(b) $9.71 \text{ cm}$ to the nearest $\text{cm}$
The whole number is 9. The first decimal place is 7. Since $7 \geq 5$, we round up. The number becomes $10 \text{ cm}$.
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(c) $10.982$ to the nearest ten cents
$10.982 is $10 and 98.2 cents. To round to the nearest ten cents, we round the 98.2 cents to the nearest 10. The digit in the units place is 8, so we round up. $98.2 \approx 100$ cents. The total is then $11.00$.
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(d) $6.489 \text{ kg}$ to the nearest $\frac{1}{100} \text{ kg}$
The nearest $\frac{1}{100} \text{ kg}$ is the second decimal place. The second decimal place is 8. We look at the next digit, which is 9. Since $9 \geq 5$, we round up. The number becomes $6.49 \text{ kg}$.
Advanced Level
5. Kate says that $5192.3$ is equal to $519$ when rounded off to the nearest 10. She drops the '2' because it is less than 5. Do you agree with her? Explain your answer.
I **disagree** with Kate. To round a number to the nearest 10, you must look at the units digit, not the last digit shown. The number is $5192.3$.
The units digit is 2. Since $2 < 5$, the number should be rounded **down** to the nearest 10. The correct value is $5190$. Kate is incorrect because she dropped the units digit instead of using it to determine the correct rounding.
6. Singapore's population was $5\,077\,000$ in 2010. This value has been rounded to the nearest $1000$. What are the largest and smallest possible values of Singapore's population in 2010?
Since the population is rounded to the nearest 1000, the actual value must be in the range where it would round to $5\,077\,000$.
The rounding was done based on the hundreds digit. The midpoint for rounding up would be 500. So, any number from $5\,076\,500$ and up would round to $5\,077\,000$. The smallest possible value is $5\,076\,500$.
The midpoint for rounding down to the next thousand would be 500 less than the next thousand (5,078,000). The value would be $5\,077\,500$. So, any number up to (but not including) $5\,077\,500$ would round down. The largest possible value is $5\,077\,499$.
7. Farhan says that $26.97$ is equal to $27$ when rounded off to 1 decimal place because he thinks that $27.0$ is the same as $27$. Do you agree with him? Explain your answer.
I **disagree** with Farhan. While the numerical value of $27.0$ is the same as $27$, the **precision** is different. When a number is rounded to 1 decimal place, it should be written with one digit after the decimal point to show that level of precision.
To round $26.97$ to 1 decimal place, we look at the second decimal place, which is 7. Since $7 \geq 5$, we round up. This causes a cascade, rounding the 9 up to a 10 and carrying the 1 over. The number becomes 27.0.
The answer is **$27.0$**, not $27$. The trailing zero is significant and indicates that the number has been measured or rounded to the tenths place, which is a more precise value than a whole number.