Question 94·Easy·Linear Inequalities in One or Two Variables
Maya’s mobile data plan allows her to use at most megabytes of data in a day. She plans to download songs, each of which is megabytes.
What is the greatest whole number of songs Maya can download without exceeding her daily data limit?
For "at most" or "at least" word problems, always start by defining a variable for the unknown quantity, then convert the words into an inequality: (amount per item) × (number of items) ≤ or ≥ (limit). Solve the inequality by performing the same operation on both sides—usually division—and then interpret your solution in context, making sure you choose a whole number if the situation requires it. Finally, quickly check your choice by multiplying back to be sure it does not violate the given limit.
Hints
Represent the unknown
Let a variable (for example, ) represent the number of songs Maya will download. How much data does she use in total if each song is megabytes?
Use the phrase "at most" correctly
"At most 450 megabytes" means the total data used can be less than or equal to . How would you write that as an inequality using and your variable?
Solve and interpret the inequality
After you write the inequality, isolate the variable by dividing both sides by . Remember that the answer must be a whole number, and you want the greatest whole number that still satisfies the inequality.
Desmos Guide
Use Desmos to compute the maximum number of songs
In Desmos, type 450/30 and press Enter. The result tells you how many 30-megabyte chunks fit into 450 megabytes. Because the number of songs must be a whole number and cannot exceed the plan's limit, use this value as the largest allowed number of songs.
Step-by-step Explanation
Define the variable
Let be the number of songs Maya downloads in a day.
Each song uses megabytes of data.
Translate the situation into an inequality
Maya can use at most megabytes. "At most" means the amount can be less than or equal to .
Total data used is megabytes per song times songs, so we write the inequality:
Solve the inequality for the number of songs
To find the largest possible , solve the inequality by dividing both sides by :
This tells us that can be any number less than or equal to , but must be a whole number because you cannot download a fraction of a song.
Choose the greatest whole number that fits
From , the greatest whole number can be is .
So, the greatest whole number of songs Maya can download without exceeding her daily data limit is 15, which corresponds to choice C.