Evaluate Expressions by Substituting Numbers | Grade 6 Math | LevelKite

A bowling alley charges 3 dollars for each game and 5 dollars to rent shoes. Let $g$ be the number of games you play. Then your total cost, in dollars, is $3g + 5$ . The term $3g$ means $3 \times g$ : 3 dollars for each of the $g$ games.

Here $g$ is a variable: a letter that stands for a number. So $3g + 5$ is not one number yet. It is a rule that gives the cost as soon as you know $g$ .

To evaluate an expression means to find the single number it equals when its variable has a given value. That number is the value of the expression. You find it by substituting: you replace the variable with the given number, and then you calculate.

The table evaluates $3g + 5$ for 1 game, 2 games and 4 games.

The cost of bowling for 1, 2 and 4 games

Games played	Substitute for the variable	After multiplying	Cost in dollars
$g = 1$	$3 \times 1 + 5$	$3 + 5$	$8$
$g = 2$	$3 \times 2 + 5$	$6 + 5$	$11$
$g = 4$	$3 \times 4 + 5$	$12 + 5$	$17$

Key concept

Evaluate by substituting

To evaluate an expression, replace the variable with the given number wherever the variable appears. Then calculate until one number is left. The expression stays the same. Its value depends on the number you substitute.

Two facts guide the calculation:

A number written directly in front of a variable multiplies it. So $3g$ becomes $3 \times 4$ when $g = 4$ .
The expressions in this lesson use only $+$ , $-$ , $\times$ and $\div$ , with no parentheses. For these, multiply and divide first, from left to right. Then add and subtract, from left to right.

The given number can be a whole number, a decimal or a fraction. The steps are the same each time.

Worked example

A whole-number value

Evaluate $3n + 24 \div n$ for $n = 4$ .

Show the hidden multiplication. The term 3n means 3 times n, because a number written directly in front of a variable multiplies it. The new line means exactly the same as the old one.
$3n + 24 \div n = 3 \times n + 24 \div n$
Substitute 4 for n. The variable appears twice, and each n stands for the same number, so replace it in both places.
$3 \times 4 + 24 \div 4$
Multiply and divide before you add. The expression adds two parts, so you need the value of each part first. This expression has no parentheses, so that is the agreed order.
$3 \times 4 + 24 \div 4 = 12 + 6$
Add the two parts. You are finished when one number is left.
$12 + 6 = 18$

Answer

When $n = 4$ , the value of $3n + 24 \div n$ is $18$ .

So far, every value has been a whole number. In the bowling example, $g$ counts games, so it is a whole number. But many quantities, such as a length or a weight, are often decimals or fractions. A variable can stand for those numbers too.

The method does not change. You still substitute, and you still calculate in the same order. Only the arithmetic is different, because now you work with decimals and fractions.

The table evaluates one expression, $8n - 2$ , for a whole number, a decimal and a fraction.

One expression, three kinds of numbers

Value of the variable	Substitute	Multiply first	Then subtract
$n = 3$	$8 \times 3 - 2$	$8 \times 3 = 24$	$24 - 2 = 22$
$n = 0.5$	$8 \times 0.5 - 2$	$8 \times 0.5 = 4$	$4 - 2 = 2$
$n = \frac{3}{4}$	$8 \times \frac{3}{4} - 2$	$8 \times \frac{3}{4} = \frac{24}{4} = 6$	$6 - 2 = 4$

Worked example

A decimal value

Evaluate $2.5w - 1.2$ for $w = 0.6$ .

Substitute 0.6 for w. Write the times sign, because 2.5w means 2.5 times w.
$2.5w - 1.2 = 2.5 \times 0.6 - 1.2$
Multiply before you subtract. Since 0.6 is 6 tenths, first find 2.5 times 6, which is 15. So the product is 15 tenths, and 15 tenths is 1.5.
$2.5 \times 0.6 = 1.5$
Subtract by place value: 5 tenths minus 2 tenths is 3 tenths, and 1 one minus 1 one is 0 ones.
$1.5 - 1.2 = 0.3$

Answer

When $w = 0.6$ , the value of $2.5w - 1.2$ is $0.3$ .

Worked example

A fraction value

Evaluate $6r - \frac{1}{2}$ for $r = \frac{3}{4}$ .

Substitute three fourths for r and write the times sign. Without the sign, a 6 next to three fourths would look like the mixed number six and three fourths, which means 6 plus three fourths. The expression needs 6 times three fourths.
$6r - \frac{1}{2} = 6 \times \frac{3}{4} - \frac{1}{2}$
Multiply first. To multiply a whole number by a fraction, multiply the whole number by the numerator and keep the denominator.
$6 \times \frac{3}{4} = \frac{6 \times 3}{4} = \frac{18}{4}$
Write eighteen fourths with a denominator of 2 so that it matches one half. Dividing the numerator and the denominator by the same number, 2, gives an equal fraction.
$\frac{18}{4} = \frac{18 \div 2}{4 \div 2} = \frac{9}{2}$
Subtract. The denominators are the same, so subtract the numerators and keep the denominator.
$\frac{9}{2} - \frac{1}{2} = \frac{8}{2}$
Eight halves make 4, because 8 divided by 2 is 4.
$\frac{8}{2} = 4$

Answer

When $r = \frac{3}{4}$ , the value of $6r - \frac{1}{2}$ is $4$ .

The big idea: to evaluate an expression with one variable, substitute the given number for the variable wherever it appears, then calculate until one number is left. Write hidden multiplication with a $\times$ sign. For expressions like the ones here, with only the four operations and no parentheses, multiply and divide before you add and subtract. The method is the same for whole numbers, decimals and fractions.

Next, you will evaluate expressions that have more than one variable, and expressions with exponents and grouping symbols.