A smartphone's battery is fully charged to 100\%. When the phone is in sleep mode, the bat… — ACT

A smartphone's battery is fully charged to $100\%$ . When the phone is in sleep mode, the battery drains at a constant rate of $1\%$ per hour. When the phone is in active use, the battery drains at a constant rate of $6\%$ per hour. Over a $20$ -hour period, the phone is either in sleep mode or in active use, and the battery level decreases to $45\%$ . For how many hours was the phone in active use during this period?

A
5
7Answer
C
9
D
11
E
13

Answer

The phone was in active use for 7 hours.

To find the active hours, first calculate the total percentage of the battery consumed:

100\% - 45\% = 55\%

. Let

x

represent the number of hours in active use. The remaining time,

20 - x

, represents the hours in sleep mode. Set up the equation representing the total battery drain:

6x + 1(20 - x) = 55

. Simplifying this equation gives

5x + 20 = 55

, which reduces to

5x = 35

, resulting in

x = 7

hours.

Step-by-Step Solution

Calculate the total percentage of the battery that drained during the period.

100\% - 45\% = 55\%

To determine the exact battery percentage consumed by both active use and sleep mode combined.

Set up a system of equations using

x

for hours in active use and

y

for hours in sleep mode.

x + y = 20

and

6x + 1y = 55

To represent the constraints on the total duration (20 hours) and the total battery drain (55%).

Substitute

y = 20 - x

into the drain equation and solve for

x

6x + (20 - x) = 55 \implies 5x + 20 = 55 \implies 5x = 35 \implies x = 7

To find the number of hours the phone was actively used.

Key Concept

Translating and solving a system of linear equations from a real-world word problem.

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