Question 80·Medium·Linear Equations in Two Variables
Calorie Information for Nuts
| Nut | Calories per ounce |
|---|---|
| Almonds | 163 |
| Cashews | 157 |
| Peanuts | 161 |
| Walnuts | 185 |
A manufacturer is creating a custom trail mix that will contain only almonds, cashews, and peanuts. The finished batch is planned to contain a total of 4,000 calories. If 8 ounces of cashews are used in the mix, which of the following equations could represent the relationship between the number of ounces, , of almonds and of peanuts that are included?
For mixture or combination problems like this, first translate the table into usable numbers (calories per ounce), then handle any part of the mix that is fully known (here, the 8 ounces of cashews). Compute that known contribution and subtract it from the total to see what remains for the unknown parts. Finally, express each unknown part as (calories per ounce) × (number of ounces), use those as coefficients of the variables, and set their sum equal to the remaining calories; watch for common traps like mixing up which calorie value goes with which variable or using the wrong total on the right-hand side.
Hints
Account for the cashews first
Before writing an equation for almonds and peanuts, figure out how many calories come from the 8 ounces of cashews using their calories per ounce from the table.
Use the total calories to find what remains for almonds and peanuts
Once you know the cashew calories, subtract that amount from 4,000 to find how many calories must come from the almonds and peanuts combined.
Match coefficients to calories per ounce
In the equation, the coefficient of should be the calories per ounce of almonds, and the coefficient of should be the calories per ounce of peanuts. The constant on the right side should be the calories that almonds and peanuts must supply together, not the full 4,000 and not just the cashew calories.
Desmos Guide
Compute remaining calories after cashews
In Desmos, type 4000 - 8*157 and press Enter. The result is the number of calories that must come from almonds and peanuts together.
Check the constant term in the options
Compare the number you got in Desmos with the constant (right-hand side) in each answer choice, and eliminate any option whose right-hand side does not match this remaining-calories value.
Match coefficients to the table
From the table, note the calories per ounce for almonds and peanuts, and check which remaining answer choice uses those two numbers as the coefficients of and in the equation.
Step-by-step Explanation
Identify what x and y represent and the given information
From the problem:
- = number of ounces of almonds
- = number of ounces of peanuts
- Cashews: 8 ounces are used.
- Total calories in the mix: 4,000.
From the table:
- Almonds: 163 calories per ounce
- Cashews: 157 calories per ounce
- Peanuts: 161 calories per ounce
We want an equation that relates and using this calorie information.
Calculate calories from the known cashews
There are 8 ounces of cashews, each with 157 calories.
So, 1,256 calories come from the cashews alone.
Find how many calories must come from almonds and peanuts
The total calories in the trail mix are 4,000. Since 1,256 calories are already from cashews, the rest must come from almonds and peanuts.
So, together, the almonds and peanuts must supply 2,744 calories.
Write the equation using x and y
Calories from almonds: (163 calories per ounce times ounces).
Calories from peanuts: (161 calories per ounce times ounces).
These together must equal the 2,744 calories that are not from cashews, so the equation is
This matches answer choice B.