Question 51·Hard·Ratios, Rates, Proportional Relationships, and Units
A large energy-blend smoothie is prepared by mixing green tea, fruit juice, and protein powder in the ratio by weight. Each large smoothie contains exactly 40 grams of protein powder.
For a special order, a super-sized jug of the energy blend must weigh three times as much as a large smoothie. The customer also requests that the jug contain 45 grams more protein powder than would be found in three large smoothies combined, while the café keeps the ratio between green tea and fruit juice the same as in the original blend.
How many grams of fruit juice should the café use to prepare the jug?
For ratio-and-units problems like this, first convert the ratio into “total parts” and use any actual measurement given (like 40 grams of one component) to find grams per part and the total weight. Then carefully apply any scaling (like tripling) and extra conditions (such as “45 grams more” of a component) to adjust the amounts. Finally, work with just the relevant ingredients, keep their ratio the same, and use proportions to turn the parts back into actual grams, watching your arithmetic on the last multiply and divide steps.
Hints
Relate the ratio to actual grams
If is the ratio of green tea, fruit juice, and protein powder, and the protein powder (the “1” part) is 40 grams, how many grams are in one “part” of the mixture, and how many grams are in the entire smoothie?
Scale up to three smoothies and then to the jug
Once you know the weight of one large smoothie and how much protein it contains, what are the total weight and total protein in three large smoothies? How does this compare to the jug's weight and protein requirement?
Focus on just tea and juice in the jug
After you find how many grams of protein powder are in the jug, subtract that from the jug’s total weight. The result is the combined weight of green tea and fruit juice. How can you divide that total using the ratio?
Turn the 4:5 ratio into actual numbers
The ratio for tea and juice means there are 9 total parts. If those 9 parts together weigh the total amount of tea plus juice you just found, how many grams is one part, and then how many grams are 5 parts (the juice)?
Desmos Guide
Write one expression for the fruit juice amount
In Desmos, type the expression:
(5/9) * (3*400 - (3*40 + 45))
This expression represents of the total weight available for tea and juice (the jug’s total weight minus its protein). The numerical value that Desmos outputs is the grams of fruit juice needed.
Step-by-step Explanation
Find the total weight of one large smoothie
The ingredients are mixed in the ratio (green tea : fruit juice : protein powder).
- Total parts parts.
- The part for protein powder equals grams.
So part grams, and the whole smoothie weighs
Find the weight and protein amount for three large smoothies
Three large smoothies together have:
- Total weight:
- Protein powder:
Determine the protein and non-protein amounts in the jug
The jug must weigh three times as much as one large smoothie, so its total weight is also
The jug must contain grams more protein powder than three smoothies combined:
So the combined weight of green tea and fruit juice in the jug is
Use the 4:5 ratio to find the grams of fruit juice
For the jug, the café keeps the ratio between green tea and fruit juice the same as the original: (tea : juice).
So:
- Total parts for tea and juice parts
- These 9 parts weigh grams
Grams per part:
Fruit juice corresponds to parts, so its weight is
Therefore, the café should use 575 grams of fruit juice for the jug.