0580 · 1.9Estimation

Rounding whole numbers and decimals

0:00Six notebooks cost four pounds and ninety-five pence each. Before calculating the exact total, you can see that the cost should be close to thirty pounds. That quick estimate gives you a useful check on your answer. In this video, we will connect rounding, estimating calculations, and choosing sensible accuracy for a final result. Rounding replaces a number with a nearby value at a chosen degree of accuracy. Estimation uses approximate values to find a result that is easier to calculate.

0:26Both deliberately leave out some detail, but they preserve information that matters. The aim is not simply to make numbers shorter, but to make them useful. The only starting knowledge we need is place value. Each position represents a different size, such as thousands, hundreds, units, tenths or hundredths. Moving one place to the right makes the value of that position ten times smaller. This helps us identify which digit to keep when rounding.

0:52Consider the instruction to write five thousand seven hundred and sixty-four correct to the nearest thousand. The neighbouring multiples of one thousand are five thousand and six thousand. Their midpoint is five thousand five hundred. Our number lies above that midpoint, so the nearer choice is six thousand. We can reach the same answer directly from the digits. The thousands digit is five, so that is the position we keep.

1:13The next digit is seven, which tells us to increase the retained digit by one. We replace the remaining whole number digits with zeros, giving six thousand. The familiar rounding rule comes from choosing the nearer value. If the next digit is less than five, round down and leave the retained digit as it is. If it is five or more, round up by adding one to the retained digit.

1:34For the positive numbers used here, an exact halfway value rounds upwards. Those zeros are essential because they preserve place value. Writing six instead of six thousand would change the size of the number completely. Rounding does not mean deleting everything after the digit you keep. For whole numbers, the positions you remove must still be represented by zeros. Now round eight point three seven six to two decimal places.

1:58Decimal places count positions after the decimal point, so the second position is the hundredths column. The retained digit is seven, and the next digit is six. We increase seven to eight, giving eight point three eight. Two decimal places means accuracy to the nearest hundredth. In this example, eight point three eight is nearer than eight point three seven. If rounding gives eight point three zero, keep the final zero when two decimal places are requested.

2:21It shows that the answer has been written to the required accuracy.

Significant figures and carrying

2:25Significant figures count from the first nonzero digit rather than from the decimal point. They describe how many digits we retain starting with the most important place value. Zeros before the first nonzero digit locate the number but do not count as significant figures. Once counting has started, zeros between other digits do count. Round zero point zero zero four seven zero six to two significant figures. The first two significant digits are four and seven.

2:49The next digit is zero, so the seven remains unchanged. The answer is zero point zero zero four seven. For three significant figures, retain four, seven and the zero that follows them. The next digit is six, so increase that retained zero to one. The answer becomes zero point zero zero four seven one. The zeros before the four remain because removing them would change the number’s size. Decimal places and significant figures therefore answer different questions.

3:19Decimal places specify a position after the decimal point. Significant figures specify how many digits to retain from the first nonzero digit. Always identify which instruction you have before deciding where to round. Sometimes rounding carries into another column. To round nine point nine six to two significant figures, retain the first two nines. The next digit is six, so increasing the tenths digit causes a carry into the units column.

3:43That carry continues into the tens column, giving ten.

Estimating calculations correctly

3:46An estimate uses rounded numbers to simplify a calculation before carrying it out. When a question says to write each number correct to one significant figure, follow that instruction for every number. Then calculate with those rounded values. Simply calculating the original expression and rounding its answer does not demonstrate the requested estimation method. Estimate forty-one point three divided by the product of nine point seven nine and zero point seven six five.

4:12First, rounded to one significant figure, forty-one point three becomes forty. Next, nine point seven nine rounds to ten. Finally, zero point seven six five rounds to zero point eight. The simplified expression is forty divided by the product of ten and zero point eight. Calculate the denominator first, giving eight. Then 40 ÷ 8 gives five. Our estimate is therefore five, and the approximate sign shows that this is not an exact equality.

4:39Keeping the denominator together is important. Dividing forty by ten and then multiplying by zero point eight would give three point two. That is a different calculation because zero point eight belongs inside the divisor. Rounding changes the numbers, but it must not change the structure of the expression. Rounding and these estimation methods must work without a calculator on Paper two. Paper four allows a calculator, but estimates still help you check entries and results.

5:03An answer wildly different from five would suggest checking the decimal points and denominator. An estimate is a reasonableness check, not proof that every digit is correct.

Choosing final answer accuracy

5:13Suppose eighteen point seven kilograms is shared equally between six containers. Calculate the mass per container by dividing eighteen point seven by six. The result is three point one one six recurring, where only the six repeats. To three significant figures, retain three, one and one, then inspect the following six. Increase the final retained one to two, giving three point one two kilograms. For an answer that is not exact, Cambridge’s usual instruction is three significant figures unless the question says otherwise.

5:42Read the question and paper instructions carefully because they can specify something different. Here, the unit is part of the final answer. Context can make another degree of accuracy more suitable. A monetary answer is normally written to the nearest penny, using two decimal places for pounds. An exact count of objects should remain a whole number. A calculator display alone does not decide how many digits belong in your final answer.

6:05Avoid rounding an answer in stages. To round two point four four nine to one decimal place, inspect the hundredths digit, which is four. The correct result is two point four. Rounding first to two decimal places gives two point four five. That would then round up to two point five, which is wrong. For an ordinary calculation, retain full calculator values during the working and round only the final answer.

6:27Estimation is the deliberate exception because it asks you to simplify the starting values. In exam working, show those rounded values and the simplified calculation. This makes your method clear even before the examiner checks your final estimate. Rounding chooses a nearby value at a specified degree of accuracy. Decimal places count from the decimal point, while significant figures start at the first nonzero digit. Estimation replaces awkward numbers with manageable ones while preserving the calculation’s structure.

6:56Finally, choose an answer’s accuracy using the instruction and the context, not every digit on the display.