What is meant by Rounding Numbers?
Rounding is a simple form of estimation. It is one of the most important skills for estimating a number quickly. This is where you ’round’ a large number by expressing it in terms of the nearest unit or a specific number of places.
For example,
1754 rounded to the nearest will be:
1. Thousands is 2,000.
2. Hundreds is 1,800.
3. Tens is 1,750.
How do we Round Numbers?
If a number has to be rounded to a particular place, in the given number, we check the digit to the immediate right of the place that has to be rounded. For instance, if a number has to be rounded to the nearest hundreds, we look at the digit in the tens place in the number. Similarly, to round to the nearest tens, we check the ones place digit.
If this digit is 5 or higher, the digit to be rounded becomes one higher, and the remaining digits to the right become zeros.
If it’s 4 or less, the digit remains the same while the rest of the numbers become zeros.
For example, 5962 to the nearest 1,000 is 6,000.
What is Approximating?
We approximate a number by rounding a number such that the number is expressed as the product of a single non-zero digit and a power of 10. For the ease of expression of very large and very small numbers, an approximation is used.
How do we Approximate Large Numbers?
Let’s understand this by approximating the number 1,893,270.
- Round the number to the largest possible place value. In this case, it is millions.
- The digit to the right of the millions place digit is 8, which is greater than 5. Hence, the millions place digit will be 1 higher, making the number 2,000,000.
- After that, express it as a product of a single digit (here 2) and a power of 10.
How do we decide the Power of 10?
If you want to convert a whole number greater than 10 to a power of 10, move the decimal point to the left from its original position and place it after the only non-zero digit. The number of places the original decimal point was moved will be a power of ten. In 2,000,000, the decimal point will move by 6 places, so 2,000,000 can be written as:
\(2,000,000 =2\times 10^6\)
Example 1:
Round the radius of the sun. Write the result as a single digit multiplied by a power of ten.
Solution:
Radius of the sun = 696,340 km.
As we know, 1 km = 1000m.
696,340 km = 696,340 × 1000 = 696,340,000 meters.
696,340,000 \(\approx\) 700,000,000 Round to the nearest 100,000,000
Factor out 7 or count the decimal places by which the decimal point will be moved:
\( = 7\times 100,000,000 \)
\( = 7\times 10^8\) write 100,000,000 as a power of 10.
Hence, the radius of the sun is \(7 × 10^8\) meters.
How do we Approximate Small Numbers?
Let’s understand this by approximating the number 0.00087654.
- Round the number to the largest possible place value. In this case, it is ten thousandths.
- The digit to the right of the ten-thousandths place digit is 7, which is greater than 5. Hence, the ten-thousandths place digit will be 1 higher, making the number 0.0009.
- After that, express it as a product of a single digit (here 9) and a power of 10.
How do we decide the Power of 10 for Small Numbers?
If the number is written in decimal form, move the decimal point to the right and place it after the first non-zero digit. The number of places the decimal point has been moved will be a power of ten. The exponent of 10 in very small numbers will be negative. In 0.0009, the decimal point will move by 4 places, so 0.0009 can be written as:
\( 0.0009 =9\times 10^{-4} \)
Example 2:
Round the number 0.000000000154. Write the result as a single-digit number multiplied by a power of ten.
Solution:
0.000000000154 \(\approx\) 0.0000000002 Round to the nearest 0.0000000001
Factor out 2 or count the decimal places by which the decimal point will be moved:
\( = 2\times 0.0000000001 \)
\(= 2\times 10^{-10}\) write 0.0000000001 as a power of 10.
Hence, the answer is \(2\times 10^{-10}\)
Solved Estimating Quantities Examples
Example 1:
Round the radius of the moon. Write the result as a single-digit multiplied by a power of ten, where the radius of the moon is 1,737,400 meters.
Solution:
\(1,737,400\approx 2,000,000 \) Round to the nearest 1,000,000
\( = 2\times 1,000,000 \) Factor out 2
\(=2\times 10^{6}\) Write 1,000,000 as a power of 10.
Hence, the radius of the moon is \(2\times 10^{6}\) meters.
Example 2:
Round the number 0.00000000024 and write the result as a single digit number multiplied by a power of ten.
Solution:
\(0.00000000024\approx 0.0000000002 \) Round to the nearest 0.0000000001
\( = 2\times 0.0000000001 \) Factor out 2
\(=2\times 10^{-10}\) Write 0.0000000001 as a power of 10
Hence, the answer is \(2\times 10^{-10}\) .
Example 3:
How big is the sun (696,340,000 meters) compared to the earth (6,371,000 meters) in terms of radius?
Solution:
You are given the radius of the sun and the earth. You are asked to approximate the number of times the radius of the sun is greater than the radius of the earth.
Each number is rounded. Each result should be written as the product of a single digit and a power of ten. Then divide the sun’s radius by the earth’s radius.
Radius of the sun:
\( 696,340,000\approx 700,000,000 \) Round to the nearest 100,000,000
\( = 7\times 100,000,000 \) Factor out 7
\( =7\times 10^8 \) Write 100,000,000 as a power of 10.
Hence, the radius of the sun is 7108 meters.
Radius of the earth:
\( 6,371,000\approx 6,000,000 \) Round to the nearest 1,000,000
\( = 6\times 1,000,000 \) Factor out 6
\( =6\times 10^6 \) Write 1,000,000 as a power of 10.
Hence, the radius of the earth is \( 6\times 10^6 \) meters.
Divide the radius of the sun by the radius of the earth.
\( =\frac{7\times 10^8}{6\times 10^6} \) \( (\frac{7}{6}=1.1666\ldots \approx 1.2) \)
\( =1.2\times 10^{8-6} \) Quotient of powers property
\( =1.2\times 10^2 \) Simplify
\( =1.2\times 100 \)
\( =120 \)
As a result, the sun is roughly 120 times the size of the earth.
Expressing numbers as multiples of 10 or scientific notation is a way of writing extremely large or extremely small numbers in a way that makes them easier to read and manipulate.
A millionth is one out of a million equal parts.