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Place value is the bedrock of the whole Number strand of OCR GCSE Mathematics (J560). Before you can confidently add, multiply, round or estimate, you need to know exactly what each digit in a number is worth. This lesson covers place value in integers and decimals, ordering numbers (including negatives), rounding to a number of decimal places and to significant figures, truncation, and estimation by rounding each value to one significant figure. These skills surface on every paper, both the non-calculator Paper 1 and the calculator Papers 2 and 3, and they underpin the bounds work you will meet later in this course.
This lesson mainly builds AO1 fluency (using and applying standard techniques accurately), with a strand of AO2 reasoning when you justify why a digit has a particular value, and AO3 problem-solving in the estimation worked examples where you decide how to simplify a messy calculation.
| Term | Meaning |
|---|---|
| Place value | The value a digit has because of its position in a number |
| Integer | A whole number — positive, negative, or zero |
| Decimal | A number written using a decimal point and place value to the right of it |
| Significant figure (s.f.) | A digit that contributes to the precision of a number, counted from the first non-zero digit |
| Decimal place (d.p.) | A digit's position after the decimal point |
| Truncate | To cut a number short by removing digits, without rounding |
| Estimate | An approximate answer, usually found by rounding each value to 1 s.f. |
| Ascending / descending | Smallest-to-largest / largest-to-smallest order |
Our number system is base-10: each column is worth ten times the column to its right, and one tenth of the column to its left. That single fact governs everything in this lesson.
Write down the value of the digit 6 in the number 4,628,503.
Reading the columns from the right, the 6 sits in the hundred-thousands column.
So the 6 is worth 600,000 (six hundred thousand).
Answer: 600,000.
Common error: Writing "hundred thousands" as the answer. OCR wants the value — the number 600,000 — not the name of the column.
The same idea runs to the right of the decimal point, where the columns are tenths, hundredths, thousandths and so on.
In the number 7.3094, write down the value of the digit 9.
After the decimal point the columns are tenths (3), hundredths (0), thousandths (9), ten-thousandths (4).
The 9 is in the thousandths column, so it is worth 10009=0.009.
Answer: 0.009.
A neat check is that the digits add back to the whole number: 7+0.3+0.00+0.009+0.0004=7.3094.
To order numbers, compare them column by column starting from the highest place value. With decimals it helps to give every number the same number of decimal places first by writing in trailing zeros — this does not change a number's value, since 0.5=0.50=0.500.
Write these numbers in ascending order: 0.61, 0.6, 0.069, 0.601.
Pad to three decimal places: 0.610, 0.600, 0.069, 0.601.
Now compare as if they were the whole numbers 610, 600, 69, 601. In order: 69,600,601,610.
Answer: 0.069,0.6,0.601,0.61.
Arrange these temperatures from coldest to warmest: −3, 5, −11, 0, −1 (in °C).
On a number line, values increase from left to right, and a negative further from zero is smaller. The coldest is the most negative.
Answer: −11,−3,−1,0,5.
Common error: Treating −11 as larger than −3 "because 11 is bigger than 3". With negatives, the larger the digit, the colder (smaller) the value.
OCR Exam Tip: When a question mixes positive and negative decimals, write them all to the same number of decimal places, sketch a quick number line, and place the negatives to the left of zero. The command word here is usually "Write down" or "Order".
To round to a given number of decimal places:
Work out 5.7468 rounded to 2 decimal places.
Two decimal places gives 5.74… The decider (third decimal) is 6, which is ≥5, so round up.
Answer: 5.75.
Significant figures are counted from the first non-zero digit. Leading zeros are never significant; zeros sandwiched between non-zero digits always are.
| Number | Significant figures |
|---|---|
| 508 | 3 (the middle zero counts) |
| 0.00420 | 3 (the leading zeros do not count; the trailing zero does) |
| 63,000 | at least 2 (trailing zeros in an integer are ambiguous) |
Round 0.0073812 to 2 significant figures.
The first significant figure is 7, the second is 3. The decider is the next digit, 8, which is ≥5, so round up the 3 to a 4.
Answer: 0.0074.
Round 48,615 to 3 significant figures.
The first three significant figures are 4, 8, 6. The decider is the next digit, 1 (<5), so leave the 6 unchanged. Replace the remaining digits with zeros to hold place value.
Answer: 48,600.
Common error: Writing 486 instead of 48,600. You must keep the place-holding zeros, or the number changes from forty-eight thousand to four hundred and eighty-six.
To truncate a number, you simply chop off the unwanted digits — you do not round.
Truncate 9.2867 to 2 decimal places.
Keep the first two decimals and discard the rest: 9.28. We do not look at the next digit at all.
Answer: 9.28.
Contrast this with rounding 9.2867 to 2 d.p., which gives 9.29 (because the decider 6≥5). Truncation always rounds towards the lower value, so it can only ever stay the same or go down.
To estimate the answer to a calculation, round every number to 1 significant figure, then work out the simpler sum. This is a guaranteed Paper 1 skill.
Work out an estimate for 0.2159.3×4.8.
Round each value to 1 s.f.: 59.3≈60, 4.8≈5, 0.21≈0.2.
0.260×5=0.2300=1500.
Answer: approximately 1500. (A calculator gives ≈1355, so the estimate is sensible.)
OCR Exam Tip: Show each "≈" rounding explicitly — those lines earn the method marks even if your final arithmetic slips. Dividing by a number smaller than 1, such as 0.2, increases the result, which catches many candidates out.
Round 4.7961 to 2 decimal places.
The first two decimals are 7 and 9, giving 4.79… The decider is 6 (≥5), so round up. Rounding the 9 in the hundredths column up makes it 10, which carries: 4.79+0.01=4.80.
Answer: 4.80 — and the trailing zero must stay to show the answer is correct to 2 d.p.
Work out an estimate for 0.4996×31.4.
96≈100=10, 31.4≈30, 0.49≈0.5.
0.510×30=0.5300=600.
Answer: approximately 600.
A bill of £47.3826 is to be rounded to the nearest penny. Work out the rounded amount.
The nearest penny means 2 decimal places. The first two decimals are 3 and 8; the decider is 2 (<5), so leave the 8 unchanged.
Answer: £47.38.
Specimen question modelled on the OCR J560 paper format: Work out an estimate for the value of 0.0096812×0.038. Give a reason why your answer is only an estimate.
Grades 3–4 response: 812≈800, 0.038≈0.04, 0.0096≈0.01. So 0.01800×0.04=0.0132=3200. It is an estimate because I rounded the numbers.
Grades 5–6 response: Rounding each value to 1 s.f.: 812≈800, 0.038≈0.04, 0.0096≈0.01. Then 0.01800×0.04=0.0132=3200. The answer is an estimate because each number was rounded to 1 significant figure before calculating, so the result is approximate rather than exact.
Grades 7–9 response: Round to 1 s.f.: 0.01800×0.04. The numerator is 800×0.04=32; dividing by 0.01 multiplies by 100, giving 3200. It is an estimate because rounding each factor introduces error; rounding 0.0096 up to 0.01 enlarges the denominator, which pulls the estimate below the true value, so I would expect the exact answer to be a little above 3200. (The calculator value is ≈3214, confirming the direction of the rounding error.)
Examiner-style commentary: the Grades 3–4 answer scores the method and a bare reason; the Grades 5–6 answer states the "1 s.f." rule explicitly; the Grades 7–9 answer reasons about the direction of the rounding error, which is the AO2 discriminator.
Specimen question modelled on the OCR J560 paper format: Write the numbers −0.4, 31, −21, 0.34 in order, starting with the smallest. Give a reason for the position of 31.
Grades 3–4 response: 31=0.33 and −21=−0.5. In order: −0.5,−0.4,0.33,0.34, so −21,−0.4,31,0.34.
Grades 5–6 response: Convert all to decimals: −0.4, 31=0.33˙, −21=−0.5, 0.34. Smallest first: −0.5,−0.4,0.333…,0.34. So the order is −21,−0.4,31,0.34. 31 comes before 0.34 because 0.333…<0.34.
Grades 7–9 response: Writing each as a decimal: −21=−0.5, −0.4, 31=0.33˙, 0.34. Ordering ascending: −0.5,−0.4,0.33˙,0.34. 31 sits just below 0.34: comparing to three decimal places, 0.333…<0.340, and the gap is only about 0.0067. The negatives are ordered "in reverse" of their digits — −0.5 is smaller than −0.4 because it is further from zero.
Examiner-style commentary: converting everything to a common form (decimals) is the key move; the top band quantifies why 31 and 0.34 are close and handles the negative ordering with confidence.
Round 26,748 to (a) the nearest 10, (b) the nearest 100, (c) the nearest 1000.
For each, find the column you are rounding to and use the next digit as the decider.
(a) Nearest 10: the tens digit is 4, the decider (units) is 8≥5, so round the 4 up to 5 and fill the units with a zero: 26,750.
(b) Nearest 100: the hundreds digit is 7, the decider (tens) is 4<5, so leave the 7 and replace the lower digits with zeros: 26,700.
(c) Nearest 1000: the thousands digit is 6, the decider (hundreds) is 7≥5, so round up to 7: 27,000.
Answers: (a) 26,750, (b) 26,700, (c) 27,000.
Common error: Rounding the same number "in stages" — first to the nearest 10, then using that answer to round to the nearest 100. Always round the original number directly, or you can drift to the wrong value.
Five rivers have lengths 346 km, 34.6 km, 364 km, 3460 m and 0.346 km. Write them in order, shortest first.
The trap is the mixed units: 3460 m must be converted to kilometres first, 3460 m=3.46 km. Now compare like with like (all in km): 0.346, 3.46, 34.6, 346, 364.
Answer: 0.346 km, 3460 m, 34.6 km, 346 km, 364 km.
This shows why "compare from the highest place value" only works once every quantity is in the same unit — converting first is the essential first step whenever units are mixed.
Specimen question modelled on the OCR J560 paper format: The number of people at a festival is reported as 48,000, correct to 2 significant figures. Show that the number rounded to the nearest 1000 could be different from 48,000, and explain your reasoning.
Grades 3–4 response: 48,000 to 2 s.f. means the first two figures 4 and 8 are fixed. The real number could be something like 47,600, which rounds to 48,000 to 2 s.f. but to 48,000 to the nearest thousand too, or 48,400. So it could be different.
Grades 5–6 response: To 2 s.f., any value from 47,500 up to just under 48,500 rounds to 48,000. For example 47,500 rounds to 48,000 (2 s.f.) but to 48,000 to the nearest 1000 as well; however 47,501 rounds to 48,000 to 2 s.f. yet to 48,000 to the nearest thousand. A value like 48,499 rounds to 48,000 to 2 s.f. but to 48,000 to the nearest thousand — so the two accuracies usually agree, but the interval of possible values is wider for 2 s.f.
Grades 7–9 response: "48,000 to 2 s.f." fixes only the first two significant figures, so the true value lies in 47,500≤n<48,500. Within that interval, values such as 47,600 round to 48,000 to the nearest 1000, but a value like 47,500 rounds to 48,000 to 2 s.f. while 47,499 would round to 47,000. The key point is that "2 s.f." and "nearest 1000" define different rounding intervals: 2 s.f. allows anything in a 1000-wide band, so the underlying number is not pinned down to a single nearest-thousand value. Stating the degree of accuracy therefore matters — it tells the reader how much the figure can be trusted.
Examiner-style commentary: the discriminator is recognising that a stated accuracy defines an interval of possible true values, not a single number; the top band states that interval precisely and links it to why degree of accuracy must always be reported.
Place value, rounding and estimation are the silent workhorses of science and finance. In Physics and Chemistry, every measured quantity carries a degree of accuracy, and results are quoted to a sensible number of significant figures — you would never report a mass as 2.7384910 g from a balance reading 2.7 g. Estimation to 1 s.f. is exactly how scientists do a "back-of-the-envelope" sanity check before trusting a calculator. In personal finance, rounding to the nearest penny governs every bill, while rounding to 2 or 3 s.f. is how budgets, populations and national statistics are reported in the news. The same skills also feed directly into the Bounds and Error Intervals lesson later in this course, where "correct to 1 d.p." becomes a precise statement about the range a true value can take.
This content is aligned with the OCR GCSE Mathematics (J560) specification.