Question

The only gift certificates that a certain store sold yesterday were worth either 100 each or 10 each. If the store sold a total of 20 gift certificates yesterday, how many gift certificates worth \$10 each did the store sell yesterday?
(1) The gift certificates sold by the store yesterday were worth a total of between 1,650 and 1,800.
(2) Yesterday the store sold more than 15 gift certificates worth 100 each.

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
• C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient.
• E. Statements (1) and (2) TOGETHER are not sufficient

Hint:

When solving problems with multiple conditions, sometimes combining the statements allows you to narrow down the solution, but each statement alone might not be enough.

Answer 1

The Correct Option is C

Step 1: Analyze statement (1).
Let the number of \$100 certificates sold be \( x \), and the number of \$10 certificates sold be \( 20 - x \). The total value of the certificates sold is: \[ 100x + 10(20 - x) = 100x + 200 - 10x = 90x + 200 \] Statement (1) tells us that the total value of the certificates sold is between \$1,650 and \$1,800. This gives the inequality: \[ 1650 \leq 90x + 200 \leq 1800 \] Subtract 200 from both sides: \[ 1450 \leq 90x \leq 1600 \] Divide by 90: \[ \frac{1450}{90} \leq x \leq \frac{1600}{90} \quad \implies \quad 16.1 \leq x \leq 17.8 \] Since \( x \) must be an integer, \( x = 17 \). Therefore, the number of \$10 certificates sold is \( 20 - 17 = 3 \).
Thus, statement (1) alone is sufficient.
Step 2: Analyze statement (2).
Statement (2) tells us that more than 15 certificates worth \$100 each were sold. Thus, \( x>15 \). Since the total number of certificates sold is 20, the number of \$10 certificates sold is \( 20 - x \). If \( x = 17 \), then \( 20 - x = 3 \).
Thus, statement (2) alone is sufficient.
\[ \boxed{C} \]