What is an arithmetic expression?

A single number like $7$ is an expression. So is $3 + 4$. So is $5 \times 6 - 2 \div 1$. Any combination of numbers and the operations $+$, $-$, $\times$, $\div$ that, when evaluated, gives a single number, is called an arithmetic expression.

Idea

An expression is built from numbers (the operands) and operations ($+$, $-$, $\times$, $\div$). Some examples:

• $25$ is an expression. Its value is $25$.
• $7 + 3$ is an expression. Its value is $10$.
• $4 \times 5 + 2$ is an expression. Its value depends on the order in which we apply the operations.

When you read $4 \times 5 + 2$ aloud as "four times five, plus two", the comma hints which operation happens first. In written form, we use brackets to make this clear: $(4 \times 5) + 2 = 22$.

We split an expression into terms. A term is a chunk separated by $+$ or $-$ signs. For example, in $4 \times 5 + 7 - 2 \times 3$, the terms are $4\times 5$, $+7$, and $-2\times 3$. Inside each term, only $\times$ and $\div$ appear. Once each term is evaluated, you simply add or subtract them together.

Why care about terms? Because terms can be added in any order (commutative law). So $4\times 5 + 7 - 2\times 3 = 20 + 7 - 6 = 21$, and the same answer is reached as $7 + 20 - 6$ or $-6 + 7 + 20$.

A small warning about signs: when you re-order, the sign moves with its term. So $7 - 2\times 3$ is "$+7$ and $-2\times 3$", giving $7 - 6 = 1$, not $7$ followed by $6$.

Worked examples

Example 1. Identify the terms in $8 + 6 \times 3 - 4$.

Terms are pieces between $+/-$. So they are $8$, $+6\times 3$, and $-4$. Each term is evaluated separately: $8$, $18$, $-4$. Sum: $8+18-4 = 22$.

Example 2. Read aloud and find the value of $20 \div 4 + 6$.

Read as "twenty divided by four, plus six". Terms: $20\div 4 = 5$ and $+6$. Value: $5 + 6 = 11$.

Example 3. Rearrange the terms of $10 - 3 + 7 - 5$ so that you add positives first and then subtract.

Sign-with-term: $+10$, $-3$, $+7$, $-5$. Re-grouping: $(10+7) - (3+5) = 17 - 8 = 9$. Confirm: $10-3+7-5 = 7+7-5 = 9$. ✓

Example 4. A shopkeeper's expression for daily earnings is $5 \times 40 + 3 \times 60 - 50$ (in rupees). What does each term mean and what is the total?

Term $5 \times 40 = 200$ might mean "five items at $\rupee 40$". Term $3 \times 60 = 180$ might mean "three items at $\rupee 60$". Term $-50$ is some expense. Total: $200 + 180 - 50 = \rupee 330$.

Try it yourself

1. List the terms of $9 - 4 \times 2 + 7$ and compute the value.
2. Rearrange $15 + 8 - 3 + 2$ into positives-first then negatives. What is the sum?
3. Read aloud and evaluate $50 \div 10 \times 2$.
4. Identify two terms in $6 \times 5 + 4 \times 3$ and find their sum.
5. Write an expression for "double of $7$, plus triple of $5$, minus $4$". Evaluate.
6. True or false: in $8 - 5 + 3$, you can swap $5$ and $3$ to get $8 - 3 + 5$ and the value stays the same.
7. Compute $100 - 25 \times 2 + 10$.
8. Write your daily school-bag expression: "$2$ books at $\rupee 60$ plus $3$ notebooks at $\rupee 25$ minus a $\rupee 20$ pen". Find the total.

Activity

In pairs, write any expression with at least three operations on a slip. Swap slips. Each partner reads the slip aloud (with proper pauses) and computes the value. Compare answers.