7. Sarah and Kurt sold rolls of wrapping paper to raise money for a school trip. The number of rolls that Kurt sold was 20 less than 3 × the number of rolls that Sarah sold. Which of the following could be the total number of rolls that Sarah and Kurt sold?
Answer: A
164 could be the total number of rolls that Sarah and Kurt sold.
The relationship between the number of rolls sold by Sarah and Kurt can be represented mathematically. If Sarah sold \( x \) rolls, then Kurt sold \( 3x - 20 \) rolls. The total number of rolls sold is \( x + (3x - 20) = 4x - 20 \). Solving for integer values of \( x \) that make the total a valid option leads us to conclude that 164 fits this relationship.
A) 164
When we set the total number of rolls sold to 164, we have the equation \( 4x - 20 = 164 \). Solving this gives \( 4x = 184 \) and \( x = 46 \). Therefore, Sarah sold 46 rolls, and Kurt sold \( 3(46) - 20 = 138 - 20 = 118 \) rolls. The total is indeed \( 46 + 118 = 164 \), making this option valid.
B) 165
Using the same equation \( 4x - 20 = 165 \), we find \( 4x = 185 \) which results in \( x = 46.25 \). Since \( x \) must be a whole number (as rolls cannot be fractional), this option is not valid.
C) 170
For the total of 170, we set up the equation \( 4x - 20 = 170 \). This simplifies to \( 4x = 190 \) leading to \( x = 47.5 \). Again, since \( x \) must be an integer, this option is also not valid.
D) 175
Setting the total to 175 gives us \( 4x - 20 = 175 \), which simplifies to \( 4x = 195 \) and results in \( x = 48.75 \). As with the previous options, this fractional result for \( x \) means this option cannot be valid.
Conclusion
164 is the only total that satisfies the mathematical relationship between the number of rolls sold by Sarah and Kurt as integers. The other options fail because they result in non-integer values for the rolls sold, which is not feasible in this context. Thus, 164 is definitively the correct answer.