A test is worth 100 points. Each problem is worth either 2 points or 5 points. The number of 5-point problems is 22 fewer than the number of 2-point problems. How many problems of... A test is worth 100 points. Each problem is worth either 2 points or 5 points. The number of 5-point problems is 22 fewer than the number of 2-point problems. How many problems of each type are on the test?
Understand the Problem
The question is asking us to determine the quantity of two types of problems on a test worth 100 points. The points for each problem are set at either 2 or 5, with the total points derived from the quantity of each problem type. Additionally, there's a relationship between the number of 5-point problems and 2-point problems that must be taken into account.
Answer
There are \( 8 \) five-point problems and \( 30 \) two-point problems.
Answer for screen readers
There are ( 8 ) five-point problems and ( 30 ) two-point problems.
Steps to Solve
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Define Variables
Let ( x ) represent the number of 2-point problems and ( y ) represent the number of 5-point problems. -
Set Up Equations
From the problem, we know that:
- The total points scored by all problems is 100:
$$ 2x + 5y = 100 $$ - The number of 5-point problems is 22 fewer than the number of 2-point problems:
$$ y = x - 22 $$
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Substitute and Simplify
Substitute the second equation into the first equation:
$$ 2x + 5(x - 22) = 100 $$
Now simplify:
$$ 2x + 5x - 110 = 100 $$
Combine like terms:
$$ 7x - 110 = 100 $$ -
Solve for ( x )
Add 110 to both sides:
$$ 7x = 210 $$
Now divide by 7:
$$ x = 30 $$ -
Find ( y )
Use the equation ( y = x - 22 ) to find ( y ):
$$ y = 30 - 22 $$
So:
$$ y = 8 $$
There are ( 8 ) five-point problems and ( 30 ) two-point problems.
More Information
This problem illustrates a system of equations that can be solved using substitution. It's common to encounter such scenarios in algebra where relationships between quantities need to be established.
Tips
- Confusing the relationships between the variables, which can lead to setting up incorrect equations.
- Forgetting to simplify the equations properly before solving.
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