A bakery uses between 1.2 and 1.5 cups of sugar to make one batch of pastries. If the bakery produces 40 batches in a month, which inequality represents the possible total number of cups of sugar, , the bakery could use that month?
When a word problem gives a range per item (like 1.2 to 1.5 cups of sugar per batch) and then gives a total number of items (like 40 batches), first write an inequality for one item, then multiply every part of that inequality by the number of items. Finally, rewrite in terms of the variable that represents the total (here, ) and choose the answer that shows the total is between the correctly scaled lower and upper bounds.
Hints
Focus on one batch versus 40 batches
First think about how many cups of sugar are used for just one batch. How is that described in the problem?
Connect repeated batches to multiplication
If the same amount (between 1.2 and 1.5 cups) is used for each of 40 batches, what operation connects the per-batch amount to the total amount for 40 batches?
Turn the per-batch inequality into a total-amount inequality
Write an inequality that shows the sugar per batch is between 1.2 and 1.5 cups, then multiply all parts of that inequality by 40 to represent 40 batches. Which option shows that relationship for , the total sugar?
Desmos Guide
Compute the lower and upper bounds numerically
In Desmos, type 1.2*40 on one line and 1.5*40 on another line. Look at the two numerical outputs; these give the smallest and largest possible total cups of sugar for 40 batches.
Match the numerical bounds to an inequality in the choices
Compare the two numbers you got in Desmos with each answer choice. Identify the option where is written as being at least the smaller number and at most the larger number, with each bound obtained by multiplying 1.2 and 1.5 by 40.
Step-by-step Explanation
Translate the per-batch information into an inequality
The problem says the bakery uses between 1.2 and 1.5 cups of sugar to make one batch.
We can write this as an inequality for the sugar per batch (call it ):
This means each single batch uses at least cups and at most cups of sugar.
Relate one batch to 40 batches
If one batch uses cups of sugar, then 40 batches use a total of cups.
To find the range for the total , multiply each part of the per-batch inequality by 40:
This gives the lower and upper bounds for the total sugar used in 40 batches.
Rewrite in terms of the total sugar s and match the choice
We defined the total sugar as , so the middle term is just .
Replace with in the inequality from Step 2 to get:
This matches answer choice B, so is the correct inequality for the possible total cups of sugar used in the month.