C1.4, C1.5, C1.6; E1.4, E1.5, E1.6 - Decimal place value and calculations | Cambridge IGCSE Mathematics

A decimal point tells you the size of every digit, so a small change in its position can change an answer tenfold. For Core and Extended, C1.4/E1.4 develops decimal notation, C1.5/E1.5 compares values, and C1.6/E1.6 covers the four operations. You will use place value, signs and inverse operations to calculate and spot misplaced decimal points.

Decimal place value and order

A student writes $4.8\times0.6=28.8$ . Can you reject that answer before completing the multiplication? Keep the question in mind: understanding the sizes of decimal numbers can catch an error faster than repeating the calculation.

In $6.274$ , the $2$ means two tenths, the $7$ means seven hundredths, and the $4$ means four thousandths. Each place is one tenth the size of the place immediately to its left.

Units	Decimal point	Tenths	Hundredths	Thousandths
$6$	.	$2$	$7$	$4$

Decimal place value

A decimal uses places for units, tenths, hundredths and smaller powers of ten. For example,

6.274=6+\frac2{10}+\frac7{100}+\frac4{1000}.

A zero can hold a place: in $6.074$ , there are no tenths. Zeros added at the end of a decimal do not change its value, so $6.7=6.70=6.700$ .

To compare positive decimals, align the decimal points and add trailing zeros if useful. Compare the first place at which the digits differ, working from left to right. The number of digits after the point does not decide the order.

Worked example 1: comparing positive decimals

Question. Write these numbers in order of size, starting with the smallest:

6.07,\quad 6.7,\quad 6.007.

Plan. Give each number three decimal places, then compare equal-sized units.

\begin{aligned} 6.07&=6.070 && \text{add a trailing zero}\\ 6.7&=6.700 && \text{add two trailing zeros}\\ 6.007&<6.070<6.700 && \text{compare thousandths after six} \end{aligned}

Answer. $6.007,\ 6.07,\ 6.7$ .

Check. Their amounts above six are seven thousandths, seventy thousandths and seven hundred thousandths. Those amounts increase in that order.

Now change every sign to negative. Their distances from zero stay the same, but the order reverses.

Worked example 2: comparing negative decimals

Question. Write these numbers in order of size, starting with the smallest:

-6.07,\quad -6.7,\quad -6.007.

Plan. Compare their distances from zero; the negative number farther from zero is smaller.

\begin{aligned} 6.700&>6.070>6.007 && \text{compare the positive sizes}\\ -6.700&<-6.070<-6.007 && \text{reverse on the negative side} \end{aligned}

Answer. $-6.7,\ -6.07,\ -6.007$ .

Check. Start at $-6.7$ on a number line. Moving $0.63$ right reaches $-6.07$ ; another $0.063$ right reaches $-6.007$ . The values increase.

The comparison symbols keep their usual meanings: $0.50=0.5$ , $0.50\ge0.5$ and $0.50\le0.5$ are true, but $0.50\ne0.5$ is false.

Watch out

$0.9<0.12$ is wrong. Nine tenths is $0.90$ , which is greater than twelve hundredths. Compare place values, not the whole numbers formed by deleting the decimal points.

Adding and subtracting decimals

Addition and subtraction combine quantities of the same size: tenths with tenths, hundredths with hundredths. Align the decimal points, not the last digits. Add trailing zeros to make any empty places visible.

For subtraction, you can exchange one unit for ten tenths, or one tenth for ten hundredths. This is the same regrouping used with whole numbers. With negative decimals, retain the number-line rules: $-2.4+0.7=-1.7$ , while $-2.4-0.7=-3.1$ .

Worked example 3: adding different numbers of decimal places

Question. Work out $18.6+2.75$ .

Plan. Write $18.6$ as $18.60$ , then add corresponding places.

\begin{aligned} 18.60+2.75 &=18+2+0.60+0.75 && \text{separate units and fractions}\\ &=20+1.35 && \text{add hundredths}\\ &=21.35 && \text{combine} \end{aligned}

The fractional total $135$ hundredths includes one extra unit and $35$ hundredths. In column addition, that is the carry into the units column.

Answer. $21.35$ .

Check. Undo the addition: $21.35-2.75=18.60$ . Also, adding a number between two and three to $18.6$ must give an answer between $20.6$ and $21.6$ .

Keep the numbers and change only addition to subtraction. The place-value alignment stays the same; the regrouping changes.

Worked example 4: subtracting with regrouping

Question. Work out $18.6-2.75$ .

Plan. Work in hundredths so that both amounts use the same-sized pieces.

\begin{aligned} 18.60-2.75&=\frac{1860-275}{100} && \text{express both in hundredths}\\ &=\frac{1585}{100} && \text{subtract whole-number counts}\\ &=15.85 && \text{convert hundredths to a decimal} \end{aligned}

This is also what a column method does: write $18.60$ , then regroup to subtract five hundredths and seven tenths.

Answer. $15.85$ .

Check. $15.85+2.75=18.60$ , the starting amount. Subtracting a number between two and three from $18.6$ must give a result between $15.6$ and $16.6$ .

If a subtraction crosses zero, compare the sizes first. For instance, $2.75-18.6=-(18.6-2.75)=-15.85$ .

Watch out

Writing $18.6$ as $18.06$ changes the value. The valid extra zero goes at the end: $18.6=18.60$ . Keep the decimal points underneath one another.

Multiplying and dividing decimals

For multiplication, first use whole-number multiplication, then restore the place values. In $4.8\times0.6$ , the first number is $48\div10$ and the second is $6\div10$ . The product is therefore $48\times6\div100$ . This explains the familiar rule of counting the total decimal places in the factors.

For division, multiply both numbers by the same power of ten until the divisor is a whole number. The quotient stays the same: a length contains the same number of equal pieces whether you measure everything in metres or centimetres. Dividing by zero remains undefined; zero divided by a nonzero decimal is zero.

Worked example 5: multiplying tenths

Question. Work out $4.8\times0.6$ .

Plan. Multiply $48$ by $6$ , then divide by $100$ to restore two decimal places.

\begin{aligned} 48\times6&=288 && \text{whole-number multiplication}\\ 4.8\times0.6&=288\div100 && \text{each factor was scaled by ten}\\ &=2.88 && \text{restore the scale} \end{aligned}

Answer. $2.88$ .

Check. Half of $4.8$ is $2.4$ and one tenth is $0.48$ . Six tenths gives $2.4+0.48=2.88$ .

Now keep both numbers but change multiplication to division. You are counting how many groups of size $0.6$ fit into $4.8$ , so the scaling step has a different purpose.

Worked example 6: dividing by tenths

Question. Work out $4.8\div0.6$ .

Plan. Multiply both dividend and divisor by ten, then divide whole numbers.

\begin{aligned} 4.8\div0.6&=48\div6 && \text{scale both numbers equally}\\ &=8 && \text{eight groups of six} \end{aligned}

Answer. $8$ .

Check. $8\times0.6=4.8$ . A divisor smaller than one can give a quotient greater than the dividend.

If the divisor has two decimal places, scale both numbers by $100$ . For example, $1.26\div0.03=126\div3$ . Apply the usual sign rule separately: one negative gives a negative result, and two negatives give a positive result.

Watch out

$4.8\div0.6=48\div0.6$ is wrong because only the dividend has changed. The equivalent whole-number calculation is $48\div6$ . Scaling both by ten preserves the quotient.

Checking decimal calculations

Two checks catch different mistakes. A size check uses what the operation must do; an inverse check reverses it with a different calculation. Simply repeating the same written steps can repeat the same error.

For positive numbers, multiplying by a number between zero and one makes the result smaller; multiplying by a number greater than one makes it larger. Dividing by a number between zero and one makes the result larger; dividing by a number greater than one makes it smaller. Multiplying or dividing by one leaves the value unchanged. For negative results, compare their sizes and check the sign separately.

Worked example 7: detecting a product that is too large

Question. A student writes $7.2\times0.4=28.8$ . Explain how you know the answer is wrong, then correct it.

Plan. Check the required size before calculating the product.

\begin{aligned} 0&<0.4<1 && \text{the multiplier is a proper part}\\ 0&<7.2\times0.4<7.2 && \text{the product must be smaller}\\ 72\times4&=288 && \text{calculate the digits}\\ 7.2\times0.4&=2.88 && \text{two decimal places in total} \end{aligned}

Answer. $28.8$ is too large; the correct answer is $2.88$ .

Check. $2.88\div0.4=28.8\div4=7.2$ , restoring the first factor.

Change only the multiplier from $0.4$ to $4$ . The product should now be ten times as large, so the size check changes direction.

Worked example 8: detecting a product that is too small

Question. A student writes $7.2\times4=2.88$ . Explain how you know the answer is wrong, then correct it.

Plan. Check whether multiplication by four has increased the positive value.

\begin{aligned} 4&>1 && \text{the multiplier is greater than one}\\ 7.2\times4&>7.2 && \text{four copies must be larger}\\ 72\times4&=288 && \text{whole-number product}\\ 7.2\times4&=28.8 && \text{one decimal place in total} \end{aligned}

Answer. $2.88$ is too small; the correct answer is $28.8$ .

Check. Add four copies: $7.2+7.2+7.2+7.2=14.4+14.4=28.8$ . This checks the product without reusing the decimal-place count.

For a practical answer, check the unit too. $3.5\,\mathrm{m}+0.8\,\mathrm{m}=4.3\,\mathrm{m}$ ; the unit stays metres. If you buy three identical items priced at $\$ 2.40 $, the total is$ $7.20$, with two decimal places in the money amount.

Problem-solving

First predict whether the answer should be larger or smaller in size. Then calculate and reverse the operation. Agreement between those checks is stronger evidence than the same calculation performed twice.

Quick Recap

Decimal places represent tenths, hundredths, thousandths and smaller equal parts.
Align decimal points for addition and subtraction. Add trailing zeros when useful.
Restore the factors' combined scale after multiplying. For division, scale both numbers equally.
Check the sign, the expected size and the inverse operation.

The opening answer $28.8$ was too large: multiplying $4.8$ by $0.6$ must make it smaller. The correct product is $2.88$ .

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C1.4, C1.5, C1.6; E1.4, E1.5, E1.6 - Decimal place value and calculations

Decimal place value and order

Adding and subtracting decimals

Multiplying and dividing decimals

Checking decimal calculations

Quick Recap

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