Question 96·Hard·Linear Equations in One Variable
A factory produced gadgets over two days. On the second day, it produced 80 fewer than 1.5 times the number of gadgets produced on the first day. In total, the factory produced 520 gadgets over the two days. How many gadgets were produced on the first day?
(Express the answer as an integer)
For word problems that describe quantities over time, first choose a clear variable (like for the first day) and rewrite each phrase carefully in algebraic form—pay special attention to comparison words such as "times" (multiply) and "fewer than" (subtract, in the correct order). Then add the expressions if a total is given, form a single linear equation, combine like terms, and solve step by step, clearing decimals if needed to keep arithmetic simple. Finally, check by plugging your answer back into the original story to see if both days add to the stated total.
Hints
Set up a variable
Let be the number of gadgets produced on the first day. Keep everything in terms of .
Translate the second-day description
"1.5 times the number on the first day" is . How do you write "80 fewer than" that amount?
Use the total to form an equation
Add your expression for the first day and your expression for the second day, and set that sum equal to 520.
Solve carefully
After combining like terms, isolate step by step. If decimals bother you, you can clear them by multiplying the whole equation by 10.
Desmos Guide
Enter expressions for each side of the equation
In one line, type y = x + 1.5x - 80. In another line, type y = 520. These represent the total gadgets from your expression and the actual total.
Find the intersection point
Look for the point where the two graphs intersect. Tap or click that intersection; note the x-value of this point. That x-value is the number of gadgets produced on the first day.
Step-by-step Explanation
Define a variable and write the second-day expression
Let be the number of gadgets produced on the first day.
"1.5 times the number of gadgets produced on the first day" is .
"80 fewer than 1.5 times" means we subtract 80 from that amount, so the number of gadgets produced on the second day is:
Use the total number of gadgets to write an equation
The total number of gadgets over the two days is 520.
Total (first day) (second day), so:
Now combine like terms on the left side:
Solve the linear equation for x
Start with:
Add 80 to both sides:
So we have:
Divide both sides by 2.5:
To divide by a decimal, multiply top and bottom by 10:
Compute the value and interpret it
Now divide:
So .
This means the factory produced 240 gadgets on the first day.