FN2–FN3, HN2–HN3 - Calculating with decimals | Eduqas GCSE Mathematics

Why is $4.62\div0.06$ larger than $4.62\div0.6$ ? Learn to calculate with decimal place values, scale multiplication and division correctly, and check signed answers using their size and inverse operations. This develops decimal arithmetic in FN2 and HN2, with the inverse-operation reasoning in FN3 and HN3.

Decimal addition and subtraction

Decimal addition and subtraction work by combining matching place values: tenths with tenths, hundredths with hundredths, and so on.

The decimal point fixes the position of the ones column. Align the decimal points in written work, then add trailing zeros where needed. Thus $8.07=8.070$ : both are the same value, now written in thousandths. This makes $8.07$ and $2.685$ comparable column by column.

You already know

Each column is worth ten times the column to its right. Regrouping one unit in a column gives ten units in the next column. A trailing zero changes no value; inserting a zero between existing digits can change it.

Once the places align, the carrying and borrowing are the same as in integer arithmetic. Keep the decimal point in that same column in the answer.

Worked example 1: adding decimals of different lengths

Question. Work out $8.07+2.685$ .

Plan. Write $8.07$ as $8.070$ , align the points, then add matching columns.

\begin{array}{r} 8.070\\ +\,2.685\\ \hline 10.755 \end{array}

The thousandths give $0+5=5$ . The hundredths give $7+8=15$ : write $5$ hundredths and carry $1$ tenth. The tenths then give $0+6+1=7$ , and the whole numbers give $8+2=10$ .

Answer. $10.755$ .

Check. Add in pieces: $8.07+2=10.07$ , then $+0.6=10.67$ , then $+0.085=10.755$ .

Change addition to subtraction. The alignment stays the same; now regroup when a column has too few units to subtract.

Worked example 2: subtracting with regrouping

Question. Work out $8.07-2.685$ .

Plan. Align to thousandths, use the usual borrowing method, and keep the answer in thousandths.

\begin{array}{r} 8.070\\ -\,2.685\\ \hline 5.385 \end{array}

Borrow one hundredth for the thousandths column: $10-5=5$ . The hundredths column now has $6$ , which is too few to subtract $8$ . Regroup one of the $8$ wholes into ten tenths, then one tenth into ten hundredths. That leaves $7$ wholes, $9$ tenths and $16$ hundredths. The remaining subtraction gives $16-8=8$ hundredths, $9-6=3$ tenths and $7-2=5$ wholes.

Answer. $5.385$ .

Check. The inverse addition gives $5.385+2.685=8.070$ , the original value.

If you reverse this subtraction, the result is negative: $2.685-8.07=-5.385$ . Find the positive difference and then use the direction of the subtraction to choose its sign. For addition with a negative, subtract its magnitude: $8.07+(-2.685)=5.385$ . Subtracting that negative instead gives $8.07-(-2.685)=10.755$ .

Watch out

Do not line up the final digits of $8.07$ and $2.685$ . The $7$ counts hundredths and must sit over the $8$ , not over the $5$ . Align decimal points first.

Decide the sign before doing the written subtraction.

Decimal multiplication

Decimal multiplication can be reduced to integer multiplication, provided you undo all the scaling afterwards.

For $2.4\times0.35$ , multiplying the first factor by $10$ gives $24$ , and multiplying the second by $100$ gives $35$ . The new product is $10\times100=1000$ times the original product. Divide the integer product by $1000$ to restore its value.

This explains the usual shortcut of counting decimal places. It works because each decimal place represents a factor of $10$ ; the total scaling depends on both factors. Use the numerical size as well as the shortcut to check the answer.

Worked example 3: restoring the product's scale

Question. Work out $2.4\times0.35$ .

Plan. Calculate $24\times35$ using integer place values, then divide by $1000$ .

\begin{aligned} 24\times35&=24\times30+24\times5 &&\text{split }35\text{ into tens and ones}\\ &=720+120=840\\ 2.4\times0.35&=840\div1000 &&\text{undo factors }10\text{ and }100\\ &=0.84 \end{aligned}

Answer. $0.84$ .

Check. $0.35=0.3+0.05$ . Three tenths of $2.4$ is $0.72$ , and five hundredths is $0.12$ ; their total is $0.84$ . Also, multiplying by less than $1$ must give less than $2.4$ .

Make the second factor ten times smaller. The integer multiplication is unchanged, but the restoring scale gains another factor of $10$ .

Worked example 4: a smaller decimal factor

Question. Work out $2.4\times0.035$ .

Plan. Use $24\times35=840$ , then undo a total scale factor of $10\times1000$ .

\begin{aligned} 2.4\times0.035&=\frac{24\times35}{10\times1000} &&\text{account for both factors}\\ &=\frac{840}{10\,000}\\ &=0.084 &&\text{restore four decimal places} \end{aligned}

Answer. $0.084$ .

Check. The second factor changed from $0.35$ to $0.035$ , exactly one tenth of its old value. The product should therefore be $0.84\div10=0.084$ .

For negative values, calculate the magnitude in the same way and use the sign rule. For example, $(-2.4)\times0.035=-0.084$ , while two negative factors give a positive product. Multiplication by zero gives zero whatever the other factor is.

Watch out

$2.4\times0.035\ne0.84$ . Scaling $0.035$ to $35$ multiplies it by $1000$ , not $100$ . A result ten times too large often means one factor of $10$ was missed.

Keep track of the separate scale factor for each number.

Decimal division

Scaling the number being divided and its divisor by the same nonzero factor leaves the quotient unchanged.

The number being divided is the dividend; the number you divide by is the divisor. In $4.62\div0.6$ , they are $4.62$ and $0.6$ . Multiplying both by $10$ gives $46.2\div6$ . The dividend and each group are ten times as large, so the number of groups stays the same. This is the same equal-scaling principle used for equivalent fractions.

Make the divisor an integer, then use written division. You may append zeros to a decimal dividend when more places are needed; these zeros change no value. Record each quotient digit in its correct place.

Worked example 5: a divisor with one decimal place

Question. Work out $4.62\div0.6$ .

Plan. Multiply both numbers by $10$ , then divide $46.2$ by $6$ .

4.62\div0.6=46.2\div6.

Written-division step	Quotient contribution	Remainder
$46$ ones divided by $6$	$7$ ones, since $6\times7=42$	$4$ ones
Bring down $2$ tenths: $42$ tenths divided by $6$	$7$ tenths	$0$

The quotient is $7$ ones and $7$ tenths.

Answer. $7.7$ .

Check. Multiply by the original divisor: $7.7\times0.6=4.62$ . The quotient is greater than $4.62$ , as expected when dividing a positive number by less than $1$ .

Now make only the divisor ten times smaller. You will fit ten times as many groups into the same dividend.

Worked example 6: a divisor with two decimal places

Question. Work out $4.62\div0.06$ .

Plan. Multiply both numbers by $100$ , then divide $462$ by $6$ .

4.62\div0.06=462\div6.

Written-division step	Quotient contribution	Remainder
$46$ tens divided by $6$	$7$ tens, since $6\times7=42$	$4$ tens
Bring down $2$ ones: $42$ ones divided by $6$	$7$ ones	$0$

The quotient is $7$ tens and $7$ ones.

Answer. $77$ .

Check. $77\times0.06=4.62$ . Also, $77$ is ten times the preceding quotient $7.7$ , answering the opening question.

Signs still follow integer division: $(-4.62)\div0.6=-7.7$ , while a negative divided by a negative is positive. A zero dividend divided by a nonzero number gives zero. A zero divisor is forbidden: no number multiplied by zero can recover a nonzero dividend, and $0\div0$ has no unique quotient.

Watch out

$4.62\div0.06\ne4.62\div6$ . If you multiply the divisor by $100$ , multiply the dividend by $100$ too. Altering only the divisor changes the answer.

Check the final quotient using the original decimal divisor.

Quick Recap

Align place values for addition and subtraction, adding trailing zeros where useful.
For multiplication, undo the scale factors applied to both factors.
For division, scale dividend and divisor equally to make the divisor an integer.
Check sign and size, then use an inverse operation. A divisor ten times smaller gives a quotient ten times larger for the same dividend.

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FN2–FN3, HN2–HN3 - Calculating with decimals

Decimal addition and subtraction

Decimal multiplication

Decimal division

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