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 to answer the question asked.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
• C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient to answer the question asked.
• E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Hint:

When a Data Sufficiency problem provides a range for a value, check if only one integer solution is possible within that range. This is a common pattern for sufficiency.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This is a system of equations problem. We are asked to find the number of \$10 gift certificates sold. Let \(x\) be the number of \$100 certificates.
Let \(y\) be the number of \$10 certificates.
We are given that the total number of certificates is 20, so \(x + y = 20\).
We need to find the value of \(y\).
Step 2: Key Formula or Approach:
The total value \(V\) of the certificates sold is given by \(V = 100x + 10y\). We can express \(x\) in terms of \(y\) using the first equation: \(x = 20 - y\).
Substituting this into the value equation gives: \[ V = 100(20 - y) + 10y \] \[ V = 2000 - 100y + 10y \] \[ V = 2000 - 90y \] Step 3: Detailed Explanation:
Analyze Statement (1): The total value was between \$1,650 and \$1,800. \[ 1650<V<1800 \] Substitute our expression for V: \[ 1650<2000 - 90y<1800 \] Subtract 2000 from all parts of the inequality: \[ 1650 - 2000<-90y<1800 - 2000 \] \[ -350<-90y<-200 \] Divide all parts by -90. Remember to flip the inequality signs when dividing by a negative number: \[ \frac{-200}{-90}>y>\frac{-350}{-90} \] \[ \frac{20}{9}>y>\frac{35}{9} \] Let's write it in the standard order: \[ 3.88...>y>2.22... \] Since \(y\) must be an integer (as it's the number of certificates), the only integer value for \(y\) in this range is 3. This gives a unique value for \(y\). Therefore, statement (1) is sufficient.
Analyze Statement (2): The store sold more than 15 certificates worth \$100 each. This means \(x>15\).
Since \(x + y = 20\), we have \(x = 20 - y\). \[ 20 - y>15 \] Subtract 20 from both sides: \[ -y>-5 \] Multiply by -1 and flip the inequality sign: \[ y<5 \] This tells us that the number of \$10 certificates is less than 5. So, \(y\) could be 0, 1, 2, 3, or 4. Since there are multiple possible values for \(y\), statement (2) is not sufficient.
Step 4: Final Answer:
Statement (1) alone is sufficient to determine the number of \$10 certificates, while statement (2) alone is not.