The Four Operations and Formal Written Methods

What the four operations do

The four operations are addition, subtraction, multiplication and division. Addition combines amounts, while subtraction finds what remains or the difference between two amounts. Multiplication forms equal groups or scales a number, while division shares an amount equally or finds how many equal groups fit into it.

They form inverse pairs. Subtraction reverses addition, and division reverses multiplication. Thus, if $456+346=802$, then $802-346=456$. If $392\div8=49$, then $49\times8=392$. These inverse relationships are useful for checking answers.

The operations can be applied to integers, decimals, proper and improper fractions, and mixed numbers. This lesson focuses on the written methods; calculations involving negative-number sign rules are considered separately.

Why formal written methods work

Formal methods organise digits according to place value. In $456$, the digits represent $400+50+6$. When digits with the same place value are aligned, hundreds are combined with hundreds, tens with tens and units with units.

Pearson Edexcel may assess non-calculator arithmetic, including long multiplication and division. No particular method is prescribed, so any correct method is acceptable. Whatever method you choose, its stages should be clear and should preserve place value.

Column addition

Write the numbers underneath one another with matching place-value columns aligned. Work from right to left. If a column totals ten or more, write its units digit in that column and regroup the remaining amount into the next column.

For $456+346$:

1. Units: $6+6=12$. Write $2$ and regroup $1$ ten.
2. Tens: $5+4+1=10$. Write and regroup hundred.

Therefore $456+346=802$. The regrouped $1$ changes value as it moves: from the units column it represents one ten, and from the tens column it represents one hundred.

For decimal addition, align the decimal points:

$$\begin{aligned} 13.70\\ +\;2.58\\ \hline 16.28 \end{aligned}$$

Writing $13.7$ as $13.70$ does not change its value. The placeholder zero simply makes the hundredths column easier to see.

Column subtraction

Align the place-value columns and work from right to left. If the upper digit is smaller than the digit below it, regroup one unit from the column to the left. That unit becomes ten units in the current column.

For $573-428$, $3-8$ cannot be completed using non-negative whole numbers. Regroup one ten from the $7$ tens. This leaves $6$ tens and makes $13$ units. Then

$13-8=5,\qquad 6-2=4,\qquad 5-4=1,$

so $573-428=145$.

The same principle works across a decimal point. In $8.20-3.67$, the zero hundredths requires regrouping. Thinking of $8.20$ as $820$ hundredths gives the result $4.53$.

Check subtraction with addition: $145+428=573$.

Long multiplication with whole numbers

Long multiplication uses place-value partial products. For example,

$315\times24=315\times(20+4).$

First calculate $315\times4=1260$. Then calculate $315\times20=6300$. The second partial product begins in the tens column because the multiplier is $20$, not $2$. Add the partial products:

$1260+6300=7560.$

Therefore $315\times24=7560$.

A grid method records the same mathematics by splitting both factors into place-value parts. For example, $42\times56$ becomes $(40+2)(50+6)$, producing the partial products $2000$, $240$, and . Their total is .

Estimate before accepting an answer. Since $315\approx300$ and $24\approx20$, the product should be of the order of $6000$. An answer of $756$ would be much too small and would suggest that a place-value zero had been lost.

Multiplying decimals

A reliable method is to remove the decimal points temporarily, multiply using whole numbers and then restore the correct scale.

For $4.96\times0.7$, first calculate $496\times7=3472$. Replacing $4.96$ by $496$ multiplies it by $100$, and replacing by multiplies it by . The temporary product is therefore times too large, so

$4.96\times0.7=3.472.$

There are three decimal places altogether in the original factors, so the product has three decimal places. This is a consequence of the scaling, not an unrelated rule. An estimate confirms the position: $4.96$ is close to $5$, so multiplying by $0.7$ should give an answer close to $3.5$.

Short and long division

In $392\div8$, short division processes the dividend from left to right. Eight does not fit into $3$ hundreds, so the $3$ hundreds are regrouped as $30$ tens. Eight fits into $39$ tens four times, leaving tens. Those become units, making units. Eight fits into nine times, so

$392\div8=49.$

When the divisor is larger, write the multiplication and subtraction stages explicitly. For $756\div12$:

1. $12$ fits into $75$ six times because $6\times12=72$.
2. Subtract $72$ from $75$ to leave $$, then bring down the final to make .

Therefore $756\div12=63$. The inverse check is $63\times12=756$.

If division is not exact, the result may be expressed as a remainder, a fraction or a decimal, depending on the context. A remainder must be interpreted: if complete groups are required, part of another group may not be usable.

Dividing by a decimal

Make a decimal divisor into a whole number by multiplying both the dividend and divisor by the same power of $10$. This does not change the quotient because it creates an equivalent fraction:

$258\div0.4=\frac{258}{0.4}=\frac{2580}{4}=645.$

Multiplying only the divisor would change the calculation.

Adding and subtracting fractions

A denominator shows the size of each part. Fractions can therefore be added or subtracted directly only when they use equal-sized parts—a common denominator.

For example,

$$\frac34+\frac56 =\frac9{12}+\frac{10}{12} =\frac{19}{12} =1\frac7{12}.$$

The denominator stays $12$ because the parts are still twelfths. Adding the denominators would change the size of the parts.

For mixed numbers, first convert to improper fractions:

$$2\frac13-\frac34 =\frac73-\frac34 =\frac{28}{12}-\frac9{12} =\frac{19}{12} =1\frac7{12}.$$

Multiplying fractions

Multiply the numerators and multiply the denominators:

$\frac ab\times\frac cd=\frac{ac}{bd}.$

Common factors can be cancelled before multiplying because this replaces fractions with equivalent fractions. For example,

$$\frac23\times\frac9{10} =\frac{2\times9}{3\times10} =\frac35.$$

A whole number can be written with denominator $1$, so $5\times\frac27$ can be treated as $\frac51\times\frac27$.

Dividing fractions

To divide by a fraction, multiply by its reciprocal:

$$\frac ab\div\frac cd =\frac ab\times\frac dc.$$

Convert mixed numbers to improper fractions first. For example,

$$1\frac12\div\frac34 =\frac32\times\frac43 =2.$$

This is sensible because two groups of $\frac34$ make $1\frac12$. Only the is inverted.

Choosing and checking a method

Estimate the likely size of the answer before calculating. Afterwards:

• check addition with subtraction;
• check subtraction with addition;
• check multiplication with division;
• check division with multiplication;
• simplify fractions fully;
• compare the final value with the estimate.

Regrouping in columns, shifting a partial product when multiplying by tens and scaling both numbers when dividing by a decimal are all consequences of place value.