Math Core

Two expressions can be equal and still tell you different things. A price of $p + 0.2p$ shows the original price plus a 20% increase, while the same amount written as $1.2p$ shows that the new price is 120% of the old one. Rewriting an expression in a new form is a way to see a problem from a different angle.

Equivalent expressions

Definition

Equivalent expressions

Two expressions are equivalent if they have the same value for every value of the variable. You can turn one into the other using properties like combining like terms and the distributive property.

For example, $x + x + x + 12$ , $3x + 12$ and $3(x + 4)$ are all equivalent. Each form highlights something different:

form	what it shows
$x + x + x + 12$	three separate $x$ 's and a $12$
$3x + 12$	the total of the $x$ part and the constant part
$3(x + 4)$	three equal groups, each worth $x + 4$

Percent problems in two forms

Rewriting is especially useful with percents. Suppose a jacket costs $p$ dollars and the price goes up by 20%.

The increase is $0.2p$ , so the new price is $p + 0.2p$ . This form shows "the old price plus the increase."
Combine like terms: $p + 0.2p = 1p + 0.2p = 1.2p$ . This form shows "the new price is $1.2$ times the old price," or 120% of it.

A decrease works the same way. After a 30% discount, the price is $p - 0.3p = 0.7p$ . You pay 70% of the original price.

Percent change as one multiplication

An increase of $r$ percent: $\quad x + \dfrac{r}{100}x = \left(1 + \dfrac{r}{100}\right)x$

A decrease of $r$ percent: $\quad x - \dfrac{r}{100}x = \left(1 - \dfrac{r}{100}\right)x$

Worked example: Sales tax

A video game costs $g$ dollars, and the sales tax is 6%. Write the total cost two ways and find it when $g = 50$ .

Price plus tax: $g + 0.06g$ .
One multiplication: $1.06g$ .

When $g = 50$ : $1.06 \times 50 = 53$ . The total is $53. Check with the first form: $50 + 0.06 \times 50 = 50 + 3 = 53$ .

Worked example: What does the form tell you?

A store writes the sale price of an item as $0.85p$ , where $p$ is the original price. What is the discount?

Rewrite $0.85p$ as $p - 0.15p$ . The store takes $0.15p$ , or 15%, off the original price.

Seeing structure in a picture

A square garden has side length $s$ feet. A path of square tiles, each $1$ foot on a side, surrounds the garden, touching it on all four sides and filling the corners. How many tiles are there?

Different people count differently, and each count gives an equivalent expression:

Four sides of $s$ tiles, plus $4$ corner tiles: $4s + 4$ .
Four strips of $s + 1$ tiles each, going around the square: $4(s + 1)$ .
Two long sides of $s + 2$ tiles (they include the corners) and two short sides of $s$ tiles: $2(s + 2) + 2s$ , which simplifies to $2s + 4 + 2s = 4s + 4$ .

All three are equivalent, so they always agree. For a $10$ -foot garden, each count gives $44$ tiles.

Worked example: Perimeter two ways

A rectangle has length $\ell$ and width $w$ . Explain why $2\ell + 2w$ and $2(\ell + w)$ both give its perimeter.

$2\ell + 2w$ : add the two lengths and the two widths.
$2(\ell + w)$ : go halfway around the rectangle (one length and one width), then double it.

The distributive property shows they're equivalent: $2(\ell + w) = 2\ell + 2w$ .

Checking whether two expressions are equivalent

To check, simplify both expressions to the same form. You can also test with numbers:

If the two expressions give different values for even one number, they are not equivalent.
If they match for a couple of numbers, that's good evidence, but simplifying is what proves it.

Common mistake

Testing with just one or two values can fool you. For example, $2x$ and $x^2$ are both $0$ when $x = 0$ and both $4$ when $x = 2$ , but they are not equivalent: at $x = 3$ they give $6$ and $9$ . Simplify when you can, and test with more than one value.

Tip

When a word problem asks you to "write an expression two ways," one way usually follows the story step by step, and the other is a simplified or factored version.

Practice

Practice 1

Which expression is equivalent to $m + 0.08m$ ?

Practice 2

A shirt that costs $p$ dollars is on sale for 25% off. The sale price can be written as $kp$ . What is $k$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 3

The population of a town is $n$ people. Ten years later it is $1.4n$ . By what percent did the population increase?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 4

A dinner bill is $b$ dollars, and the family leaves an 18% tip. Which expression does not represent the total they pay?

Practice 5

Are $3(x + 2)$ and $3x + 2$ equivalent?

Practice 6

Jada earns $12 per hour at her job plus a $3 per hour bonus on weekends. Her weekend pay for $h$ hours is $12h + 3h$ . Rewrite this as a single term. How many dollars does she earn per weekend hour?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 7

Which expression is equivalent to $2(\ell + w)$ but written in a different form?

Practice 8

A square patio has side length $s$ feet. It is surrounded by a border of $1$ -foot square tiles that fills the corners, so the number of tiles is $4(s + 1)$ . How many tiles are needed when $s = 15$ ? Check your answer with the form $4s + 4$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.