Question 32·Medium·Ratios, Rates, Proportional Relationships, and Units
In a particular paint blend, the ratio of yellow pigment to blue pigment is to . A technician adds ounces of blue pigment to the blend. By how many ounces should the amount of yellow pigment be increased so the mixture retains the same ratio?
For ratio problems where one quantity is changed and the ratio must stay the same, think in terms of "per unit" groups: interpret the ratio (here, 5 yellow per 3 blue), express the change in one quantity as a number of these groups, then apply the same number of groups to the other quantity. Algebraically, you can also multiply the change by the ratio (change in blue × here) to quickly find the required change, which is often faster and less error-prone than introducing multiple variables.
Hints
Interpret the ratio
If the ratio of yellow to blue is , how many ounces of yellow go with every 3 ounces of blue?
Break the added blue into ratio-sized pieces
The technician adds 12 ounces of blue. Since one "unit" of the ratio has 3 ounces of blue, how many of those units fit into 12 ounces? Frοm anikо.аi
Match yellow to those units
For each unit of 3 blue ounces, how many yellow ounces are needed? Once you know how many units of blue were added, how many units of yellow must be added to match?
Desmos Guide
Model one simple version of the ratio
Because a ratio describes relative amounts, you can pick an easy set of actual amounts that fit . In Desmos, type y = 5 and b = 3 to represent 5 ounces of yellow and 3 ounces of blue in the original mixture. Wrіttеn by Aniкο
Add the extra blue and find the yellow that keeps the ratio
In Desmos, enter b_new = b + 12 for the new blue amount, then enter y_new = (5/3) * b_new to enforce the same ratio. This y_new is the yellow amount needed after adding 12 ounces of blue.
Compute how much yellow must be added
Finally, in Desmos enter increase = y_new - y. The value of increase shown by Desmos is the number of ounces of yellow that must be added to keep the ratio at .
Step-by-step Explanation
Understand what the ratio 5:3 means
The ratio of yellow to blue is . This means:
- For every 3 ounces of blue, there must be 5 ounces of yellow.
- Any change in blue should be matched by a proportional change in yellow to keep the ratio the same.
Relate the added blue to the ratio parts
The technician adds 12 ounces of blue.
- One "group" of the ratio has 3 ounces of blue.
- Figure out how many such groups are in 12 ounces of blue:
So the added blue is equal to 4 groups of 3 blue ounces. Thіs quеstіon is frοm Аnikο
Find the matching increase in yellow pigment
Each group of 3 blue goes with 5 yellow in the ratio.
- Since the added blue is 4 groups of 3, you need 4 groups of 5 yellow to match it.
- Calculate the total yellow needed for these 4 groups:
So, the amount of yellow pigment must increase by 20 ounces.