Number · Number and Numeration Systems · Lesson 3
Rounding Large Whole Numbers and Significant Figures
Newspapers say "Ghana has about 30.8 million people" instead of 30,832,019 because a rounded number is easier to read, remember and compare. In this lesson you will round very large whole numbers to the nearest ten, hundred, thousand, ten-thousand and hundred-thousand, learn the difference between rounding off, rounding up and rounding down, and express whole numbers to a given number of significant figures.
By the end you will be able to
- ✔Round off whole numbers (including numbers above 1,000,000,000) to the nearest ten, hundred, thousand, ten-thousand and hundred-thousand.
- ✔Explain and use the difference between rounding up, rounding down and rounding off.
- ✔Express whole numbers correct to a given number of significant figures.
- ✔Find the smallest and largest numbers that round to a given value.
Key words
- Round off
- Replace a number by the NEAREST value at the chosen place, e.g. 1,879,653 ≈ 1,900,000 to the nearest hundred-thousand.
- Round up
- Choose the LARGER of the two possible rounded values, e.g. 2,846,655 rounded up to the nearest thousand is 2,847,000.
- Round down
- Choose the SMALLER of the two possible rounded values, e.g. 2,846,655 rounded down to the nearest thousand is 2,846,000.
- Deciding digit
- The digit just to the right of the rounding place. It tells you whether to round off up or down.
- Significant figures (s.f.)
- The digits of a number, counted from the first non-zero digit, that carry its meaning. 857,000,000 is 857,386,321 to 3 s.f.
Curriculum reference
Content standard: B7.1.1.1 Demonstrate understanding and the use of place value for expressing quantities recorded as base ten numerals as well as rounding these to given decimal places and significant figures.
Indicator: Round (off, up, down) whole numbers more than 1,000,000,000 to the nearest hundred-thousand, ten-thousands, thousands, hundreds and tens.
Starter
"About 31 million Ghanaians"
The 2021 Population and Housing Census counted 30,832,019 people in Ghana. A radio presenter in Tamale says "about 31 million", a newspaper in Accra writes "30.8 million" and a textbook says "30,800,000". Are they all telling the truth? Yes — each of them has rounded the same number in a different way. By the end of this lesson you will know exactly how each one did it.
1. What rounding really means
To round a number to a chosen place, you first find the two numbers it lies between at that place. For example, to round 1,879,653 to the nearest hundred-thousand, ask: which two hundred-thousands is it between? It is between 1,800,000 and 1,900,000. Rounding off means choosing the one that is nearer.
The halfway point between 1,800,000 and 1,900,000 is 1,850,000. Our number, 1,879,653, is past the halfway point, so it is nearer to 1,900,000. Step through the number line to see this.
▶ Rounding off = choosing the nearer of the two candidates. Notice where the halfway mark is.
To the nearest ten-thousand, the same number lies between 1,870,000 and 1,880,000. The halfway point is 1,875,000 and 1,879,653 is past it, so it rounds to 1,880,000. Notice that the answer depends on the place you are rounding to.
2. The quick rule: look at the deciding digit
Drawing a number line every time is slow, so we use a rule that does the same job. The digit just to the right of the rounding place — the deciding digit — tells you whether the number is before or after halfway.
- 1Find the digit in the place you are rounding to (underline it).
- 2Look at the deciding digit, the one immediately to its right.
- 3If the deciding digit is 0, 1, 2, 3 or 4, keep the underlined digit as it is.
- 4If the deciding digit is 5, 6, 7, 8 or 9, add 1 to the underlined digit (carry if it is a 9).
- 5Replace every digit to the right of the rounding place with 0. The zeros keep the number the right size.
| Nearest | Rounding digit | Deciding digit | Answer |
|---|---|---|---|
| hundred-thousand | 5 | 8 → add 1 | 4,736,600,000 |
| ten-thousand | 8 | 2 → keep | 4,736,580,000 |
| thousand | 2 | 4 → keep | 4,736,582,000 |
| hundred | 4 | 1 → keep | 4,736,582,400 |
| ten | 1 | 9 → add 1 | 4,736,582,420 |
3. Round up, round down and round off
Every number lies between two possible rounded values. The curriculum gives three ways to choose: round up always takes the larger value, round down always takes the smaller value, and round off takes the nearer value (using the deciding-digit rule). Rounding up and down ignore the deciding digit completely.
| 2,846,655 to the nearest… | Round up | Round down | Round off |
|---|---|---|---|
| thousand | 2,847,000 | 2,846,000 | 2,847,000 |
| ten-thousand | 2,850,000 | 2,840,000 | 2,850,000 |
| hundred-thousand | 2,900,000 | 2,800,000 | 2,800,000 |
▶ Round up and round down just pick an end of the interval; round off picks the nearer end.
4. Significant figures of whole numbers
Sometimes we are not told a place to round to; instead we are told how many important digits to keep. These are the significant figures (s.f.). In a whole number, counting starts from the first non-zero digit on the left. To give 857,386,321 to 3 s.f. we keep the first three digits 8, 5, 7 and use the 4th digit as the deciding digit.
▶ Watch how only the first few digits are kept and the rest become place-holding zeros.
| Significant figures | Digits kept | Deciding digit | Answer |
|---|---|---|---|
| 5 s.f. | 8 5 7 3 8 | 6 → add 1 | 857,390,000 |
| 4 s.f. | 8 5 7 3 | 8 → add 1 | 857,400,000 |
| 3 s.f. | 8 5 7 | 3 → keep | 857,000,000 |
| 2 s.f. | 8 5 | 7 → add 1 | 860,000,000 |
| 1 s.f. | 8 | 5 → add 1 | 900,000,000 |
Worked examples
Try each one yourself first, then open the solution and compare every step.
Example 1
Nearest hundred-thousand and ten-thousand
▶ Notice how close 1,879,653 is to 1,880,000.
Show the solution
- Step 1(a) Hundred-thousands digit: 8 (in 1,879,653). Deciding digit (ten-thousands): 7, which is 5 or more, so 8 becomes 9.
- Step 2Replace the digits after it with zeros: 1,900,000.
- Step 3(b) Ten-thousands digit: 7. Deciding digit (thousands): 9, which is 5 or more, so 7 becomes 8.
- Step 4Replace the digits after it with zeros: 1,880,000.
Answer: (a) 1,900,000 (b) 1,880,000
Example 2
A number above one billion
Show the solution
- Step 1(a) Groups: 4 | 736 | 582 | 419. The thousands digit is 2; the deciding digit (hundreds) is 4, which is less than 5.
- Step 2Keep 2 and replace the last three digits with zeros: 4,736,582,000.
- Step 3(b) The tens digit is 1; the deciding digit (ones) is 9, which is 5 or more, so 1 becomes 2.
- Step 4Replace the ones digit with 0: 4,736,582,420.
Answer: (a) 4,736,582,000 (b) 4,736,582,420
Example 3
Up, down and off
Show the solution
- Step 12,846,655 lies between 2,800,000 and 2,900,000.
- Step 2(a) Round up = larger value = 2,900,000.
- Step 3(b) Round down = smaller value = 2,800,000.
- Step 4(c) Round off: the hundred-thousands digit is 8 and the deciding digit is 4 (less than 5), so it is nearer 2,800,000.
Answer: (a) 2,900,000 (b) 2,800,000 (c) 2,800,000
Example 4
Significant figures
Show the solution
- Step 1(a) Keep 8 5 7 3 8; the 6th digit is 6 (5 or more), so 8 → 9: 857,390,000.
- Step 2(b) Keep 8 5 7 3; the 5th digit is 8 (5 or more), so 3 → 4: 857,400,000.
- Step 3(c) Keep 8 5 7; the 4th digit is 3 (less than 5), so keep 7: 857,000,000.
- Step 4In each answer, fill the remaining places with zeros so the number keeps its size (nine digits).
Answer: (a) 857,390,000 (b) 857,400,000 (c) 857,000,000
Example 5
Ghana's population
Show the solution
- Step 1(a) Hundred-thousands digit: 8; deciding digit (ten-thousands): 3 < 5, so keep 8: 30,800,000.
- Step 2(b) 2 s.f.: keep 3 and 0; deciding digit 8 ≥ 5, so 0 → 1: 31,000,000.
- Step 3(c) 4 s.f.: keep 3, 0, 8, 3; deciding digit 2 < 5, so keep: 30,830,000.
- Step 4Notice the zero in 30,832,019 is significant because it lies between non-zero digits.
Answer: (a) 30,800,000 (b) 31,000,000 (c) 30,830,000
Summary
- Rounding off chooses the NEARER of the two possible values; use the deciding digit (0–4 keep, 5–9 add 1).
- Round up always chooses the larger value; round down always chooses the smaller value.
- After rounding, fill the places to the right with zeros so the number keeps its size.
- Always round from the ORIGINAL number — never round in a chain.
- Significant figures are counted from the first non-zero digit; zeros between non-zero digits count, but end zeros in a rounded whole number only hold places.
Practice: 15 questions
5 easy, 5 medium, 5 hard. Each one is marked straight away, with a full worked solution.
You can practise without an account, but sign up free or sign in to save your progress.