Question 24·Hard·Linear Equations in One Variable
A delivery truck is loaded with large packages that each weigh 18 kilograms, medium packages that each weigh 12 kilograms, and 10 small packages that each weigh 4 kilograms. The number of large packages is twice the number of medium packages. The total weight of all the packages on the truck is 460 kilograms. Which equation represents this situation?
For word problems that ask for an equation, first choose a clear variable (here, p for the number of medium packages). Rewrite all quantities in the story in terms of that variable: use the given relationships to express the counts of each item, then multiply each count by its weight or value. Add these expressions to form a single total, simplify by combining like terms, and finally set this total equal to the overall amount given in the problem. Pay close attention to avoid mixing up counts (like 10 packages) with weights (like 4 kilograms) and to avoid assigning the wrong weight to a package type.
Hints
Turn words into expressions
Focus on turning each sentence into a multiplication expression: number of packages times weight of each package for large, medium, and small packages.
Use the relationship between large and medium packages
If there are p medium packages and the number of large packages is twice that, what is the expression for the number of large packages? Then multiply that by 18 to get their total weight.
Include the small packages correctly
There are 10 small packages and each weighs 4 kilograms. What is the total weight of the small packages, and how should that appear as a constant term in the equation?
Put it all together and simplify
Add the weight expressions for large, medium, and small packages, then combine like terms (the p terms) and keep the number for the small packages as a single constant. Set this total equal to 460.
Desmos Guide
Let Desmos simplify the total weight expression
In Desmos, type the full expression for the total weight based directly on the description: 18*(2p) + 12p + 10*4. Desmos will automatically simplify this expression into a shorter form in terms of p.
Match the simplified form to an answer choice
Look at the simplified expression Desmos shows and then match that left-hand side to one of the answer choices, and mentally set it equal to 460 (the total weight) to see which equation correctly represents the situation.
Step-by-step Explanation
Define the variable and package counts
Let p be the number of medium packages.
The problem states that the number of large packages is twice the number of medium packages, so:
- Medium packages: p
- Large packages: 2p
- Small packages: 10 (this number is already given, not in terms of p).
Write an expression for the total weight
Now multiply each package count by its weight:
- Each large package weighs 18 kg and there are 2p of them, so their total weight is .
- Each medium package weighs 12 kg and there are p of them, so their total weight is .
- Each small package weighs 4 kg and there are 10 of them, so their total weight is .
Add these to get the total weight expression:
This expression must equal the total weight of 460 kg, so we will set it equal to 460 in the next step after simplifying.
Simplify and match to an equation choice
First simplify the expression for the total weight:
- Multiply for the large packages: .
- Combine the large and medium terms: .
- Multiply for the small packages: .
So the total weight becomes:
This must equal 460 kg:
This equation matches answer choice C, . This is the correct representation of the situation.