A toy company packages marbles so that for every 5 large marbles there are 3 small marbles.
If a shipment contains 120 large marbles, how many fewer small marbles than large marbles are in the shipment?
For ratio questions, first interpret the ratio as a “group” (here, 5 large and 3 small) and use the given total to find how many such groups there are by division. Then, either (1) find the difference within one group and multiply by the number of groups, or (2) find the total of the other quantity and subtract to get the requested comparison. Always re-read what the question is asking—total amount, other quantity, or a difference—to avoid stopping at an intermediate value.
Hints
Break the shipment into ratio groups
Think of the shipment as made up of repeated groups that follow the 5 large to 3 small pattern. How many large marbles are in each group?
Find the number of groups
Use the 120 large marbles and the fact that there are 5 large marbles per group to find how many full groups there are in the shipment.
Difference per group vs. whole shipment
In one group, how many more large marbles are there than small marbles? Once you know that, how can you scale this up using the number of groups?
Desmos Guide
Compute the total difference with one expression
In Desmos, type the expression 120 - (120/5*3). This uses the ratio (5 large for every 3 small) to find the total number of small marbles and then subtracts from 120. The value that Desmos outputs is the number of fewer small marbles than large marbles in the shipment.
Step-by-step Explanation
Understand the ratio and form groups
The marbles are packaged so that for every 5 large marbles there are 3 small marbles.
This means the shipment can be thought of in groups of:
- 5 large marbles
- 3 small marbles
First, find how many such groups are in 120 large marbles. Since each group has 5 large marbles, compute to get the number of groups.
Use the number of groups to compare large and small marbles
From Step 1, you found the number of groups:
- Number of groups .
In each group:
- Large marbles: 5
- Small marbles: 3
- So there are fewer small marbles than large marbles per group.
Now you know there are 24 groups, and each group has 2 fewer small marbles than large marbles.
Scale up the difference to the whole shipment
To find how many fewer small marbles than large marbles are in the entire shipment, multiply the difference per group by the number of groups:
Compute this product to get the total number of fewer small marbles than large marbles in the shipment, which is 48.