2.02b, 2.02c - Calculating with decimals | OCR GCSE Mathematics

Can dividing by a number make your answer larger? Use place value to add, subtract, multiply and divide positive and negative decimals without a calculator. You will see why decimal division changes both numbers together, and use inverse operations to check your answers.

Adding and subtracting decimals

How many $0.4$ -litre portions fit into $2.4$ litres? Keep this question for the final part. First, we need reliable ways to calculate with decimal amounts.

Adding $2.4$ and $0.35$ means adding units to units, tenths to tenths and hundredths to hundredths. Write $2.4$ as $2.40$ , so the decimal points line up. A zero at the end makes the columns visible without changing the amount.

You already know

Signed-number rules still apply to decimals. Adding a negative lowers the starting value; subtracting a negative adds its opposite. The decimal point changes the place values, not the sign rules.

You can calculate with whole numbers of hundredths, then return to the decimal amount. This explains column arithmetic, including exchanging one tenth for ten hundredths when subtracting.

Worked example 1: adding a decimal

Question. Work out $4.7+2.85$ .

Plan. Write both numbers in hundredths and add equal-sized units.

\begin{aligned} 4.7+2.85&=4.70+2.85 &&\text{align decimal places}\\ &=\frac{470+285}{100} &&\text{count hundredths}\\ &=\frac{755}{100}=7.55 &&\text{return to decimals} \end{aligned}

Answer. $7.55$ .

Check. $7.55-2.85=4.70$ . Subtracting the added amount restores the start.

Change only the sign of the second number. The aligned columns remain useful, but we now subtract its magnitude.

Worked example 2: adding a negative decimal

Question. Work out $4.7+(-2.85)$ .

Plan. Subtract $2.85$ from $4.70$ , keeping equal place values together.

\begin{aligned} 4.7+(-2.85)&=4.70-2.85 &&\text{add a negative}\\ &=\frac{470-285}{100} &&\text{count hundredths}\\ &=\frac{185}{100}=1.85 &&\text{return to decimals} \end{aligned}

Answer. $1.85$ .

Check. $1.85+2.85=4.70$ , and the answer is positive because $4.70$ is larger than $2.85$ .

The same method works for a negative starting value: $-4.7-2.85=-7.55$ . If you subtract a negative, change to addition first: $-4.7-(-2.85)=-4.7+2.85=-1.85$ .

Watch out

Do not align the last digits when adding $4.7+2.85$ . The $7$ represents tenths, so it must sit above the $8$ , not above the $5$ .

Keep the second number negative in the next sum.

Multiplying decimals

Multiplying two decimals combines their scale factors as well as their digits.

Since $0.6=\frac6{10}$ and $0.4=\frac4{10}$ , their product is $\frac{24}{100}=0.24$ . You multiply the whole-number digits, then account for each factor of ten used to make them whole numbers. For finite decimals, the two counts of decimal places add. Trailing zeros in the result may be removed.

Worked example 3: placing the decimal point

Question. Work out $1.2\times0.35$ .

Plan. Multiply $12$ by $35$ , then divide by $10\times100$ .

\begin{aligned} 12\times35&=12\times(30+5)=420 &&\text{whole-number product}\\ 1.2\times0.35&=\frac{420}{1000} &&\text{one plus two places}\\ &=0.420=0.42 &&\text{remove a trailing zero} \end{aligned}

Answer. $0.42$ .

Check. $0.35+0.2\times0.35=0.35+0.07=0.42$ . This finds one whole and two tenths of $0.35$ .

Change only the sign of the first factor. The size calculation stays the same; one negative factor gives a negative product.

Worked example 4: a negative product

Question. Work out $-1.2\times0.35$ .

Plan. Use the positive magnitude calculation, then apply the sign rule.

\begin{aligned} 1.2\times0.35&=0.42 &&\text{same magnitudes}\\ -1.2\times0.35&=-0.42 &&\text{one negative factor} \end{aligned}

Answer. $-0.42$ .

Check. The two products add to zero: $0.42+(-0.42)=0$ , matching $(1.2-1.2)\times0.35$ .

With two negative factors the product is positive: $(-1.2)\times(-0.35)=0.42$ . Multiplying a nonzero number by a positive number below one reduces its magnitude; this is why the positive product $1.2\times0.35$ is below $1.2$ .

Watch out

$1.2\times0.35\ne4.2$ . The whole-number product $420$ must be divided by $1000$ , not $100$ . Check the likely size before accepting a decimal point.

Change the first factor's magnitude and keep its negative sign.

Dividing by a whole number

When you divide a decimal by a whole number, share its units, tenths and hundredths without changing their place values.

For $7.2\div3$ , share seven units: two each, with one unit left. Exchange the remaining unit for ten tenths, combine these with the two tenths already present, then share twelve tenths: four tenths each. The answer is $2.4$ . Appending zeros allows further sharing if a remainder remains.

Worked example 5: sharing hundredths

Question. Work out $5.76\div8$ .

Plan. Treat $5.76$ as $576$ hundredths, divide, then preserve that unit.

\begin{aligned} 576\div8&=72 &&\text{whole-number division}\\ 5.76\div8&=72\text{ hundredths} &&\text{same place-value unit}\\ &=0.72 &&\text{write as a decimal} \end{aligned}

Answer. $0.72$ .

Check. $0.72\times8=5.76$ , so eight shares rebuild the total.

Change only the sign of the dividend. A negative number divided by a positive number gives a negative result.

Worked example 6: a negative dividend

Question. Work out $-5.76\div8$ .

Plan. Divide the magnitude as before, then give the quotient its sign.

\begin{aligned} 5.76\div8&=0.72 &&\text{magnitude of each share}\\ -5.76\div8&=-0.72 &&\text{opposite signs} \end{aligned}

Answer. $-0.72$ .

Check. $(-0.72)\times8=-5.76$ .

A negative divisor follows the same sign rules: $(-5.76)\div(-8)=0.72$ . Division by zero is undefined, so the divisor must be nonzero.

Watch out

$5.76\div8\ne7.2$ . The $72$ from $576\div8$ counts hundredths. Also, dividing a positive amount by $8$ must give a smaller positive amount.

Use multiplication to check the sign and size of the next quotient.

Dividing by a decimal

Dividing by $0.4$ asks how many groups of size $0.4$ fit into the amount, so the result can be larger than the starting number.

For the opening question, $2.4$ litres contains six $0.4$ -litre portions. In tenths of a litre, that is $24\div4=6$ . Changing the unit rescales both the amount and the group size by the same factor.

Equivalent division

Multiplying both the dividend and the nonzero divisor by the same nonzero number leaves the quotient unchanged. For example, $2.4\div0.4=24\div4$ . Multiplying only one of them changes the question.

Choose $10$ , $100$ or $1000$ to make the divisor a whole number. Move both numbers the same number of decimal places, even if one needs extra zeros.

Worked example 7: a divisor with two decimal places

Question. Work out $4.2\div0.35$ , without a calculator.

Plan. Multiply both numbers by $100$ to make the divisor $35$ .

\begin{aligned} 4.2\div0.35&=420\div35 &&\text{scale both by 100}\\ &=12 &&\text{since }35\times12=420 \end{aligned}

Answer. $12$ .

Check. $12\times0.35=10\times0.35+2\times0.35=3.5+0.7=4.2$ .

Change only the divisor from $0.35$ to $3.5$ . Each group is now ten times as large, so there are one tenth as many groups.

Worked example 8: a larger divisor

Question. Work out $4.2\div3.5$ , without a calculator.

Plan. Multiply both numbers by $10$ , then divide $42$ by $35$ .

\begin{aligned} 4.2\div3.5&=42\div35 &&\text{scale both by 10}\\ &=\frac{42}{35}=\frac65=1.2 &&\text{simplify the fraction} \end{aligned}

Answer. $1.2$ .

Check. $1.2\times3.5=3.5+0.7=4.2$ . The result is one tenth of $12$ , as predicted.

For signed decimals, make the magnitude calculation safe, then apply the sign: $-4.2\div0.35=-12$ . On a calculator paper, the multiplication check is still valuable; on a non-calculator paper, write the equivalent whole-number division clearly.

Watch out

$4.2\div0.35\ne4.2\div35$ . If you multiply the divisor by $100$ , multiply the dividend by $100$ too: $420\div35$ .

Make the divisor a whole number before calculating.

Quick Recap

Align place values for addition and subtraction; keep the signed-number rules.
Multiply whole-number digits, then account for both decimal scale factors.
Divide by a whole number while preserving the place-value units.
For a decimal divisor, scale both numbers equally; the divisor must be nonzero.

Dividing can increase a positive number: $2.4\div0.4=6$ , because six small portions fit into the total.

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2.02b, 2.02c - Calculating with decimals

Adding and subtracting decimals

Multiplying decimals

Dividing by a whole number

Dividing by a decimal

Quick Recap

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