Landen spent $L$ hours at the beach last weekend. Mateo spent $15\%$ fewer hours at the beach than Landen did. $\newline$ Which of the following expressions could represent how many hours Mateo spent at the beach last weekend? $\newline$ Choose $2$ answers: $\newline$ (A) $L - \frac{1}{15}L$ $\newline$ (B) $L(1 - 0.15)$ $\newline$ (C) $0.15L$ $\newline$ (D) $L - \frac{15}{100}$ $\newline$ (E) $L - \frac{3}{20}L$

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Grade 8

Identify equivalent linear expressions: word problems

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Q. Landen spent

L

hours at the beach last weekend. Mateo spent

15\%

fewer hours at the beach than Landen did.

\newline

Which of the following expressions could represent how many hours Mateo spent at the beach last weekend?

\newline

Choose

2

answers:

\newline

(A)

L - \frac{1}{15}L

\newline

(B)

L(1 - 0.15)

\newline

(C)

0.15L

\newline

(D)

L - \frac{15}{100}

\newline

(E)

L - \frac{3}{20}L

Understand the problem: Understand the problem. $\newline$ We need to find an expression that represents $15\%$ fewer hours than $L$ , where $L$ is the number of hours Landen spent at the beach.
Convert to decimal: Convert the percentage to a decimal. $\newline$ $15\%$ as a decimal is $0.15$ .
Calculate $15$ % fewer hours: Calculate the expression for $15$ % fewer hours. $\newline$ To find $15$ % fewer of $L$ , we subtract $15$ % of $L$ from $L$ . This can be written as $L - 0.15L$ .
Simplify the expression: Simplify the expression from Step $3$ . $\newline$ $L - 0.15L$ can be simplified by factoring out $L$ , which gives us $L(1 - 0.15)$ .
Convert to fraction: Convert the percentage to a fraction. $15\%$ can also be expressed as the fraction $\frac{15}{100}$ , which simplifies to $\frac{3}{20}$ .
Calculate using fraction: Calculate the expression for $15\%$ fewer hours using the fraction. $\newline$ Another way to express $15\%$ fewer of $L$ is $L - (\frac{15}{100})L$ , which simplifies to $L - (\frac{3}{20})L$ .
Check given options: Check the given options against our calculations. $\newline$ Option B, $L(1-0.15)$ , matches our expression from Step $4$ . $\newline$ Option E, $L-\left(\frac{3}{20}\right)L$ , matches our expression from Step $6$ .
Eliminate incorrect options: Eliminate incorrect options. $\newline$ Option A, $L-(\frac{1}{15})L$ , does not represent $15\%$ fewer hours. $\newline$ Option C, $0.15L$ , represents $15\%$ of $L$ , not $15\%$ fewer. $\newline$ Option D, $L-(\frac{15}{100})$ , is missing the $L$ in the second term to correctly represent $15\%$ fewer hours.