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If Jane bought 3 equally priced shirts o...

If Jane bought 3 equally priced shirts on sale, she would have 2 dollars left over. If instead she bought 10 equally priced of socks, she would have 7 dollars left over. If the prices of both shirts and socks are integers, which of the following, in dollars, could be the amount that Jane has to spends?

A

`28`

B

`32`

C

`47`

D

`57`

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to set up equations based on the information given about Jane's purchases. 1. **Let x be the price of one shirt and y be the price of one pair of socks.** According to the problem: - If Jane buys 3 shirts, she has $2 left over. This can be expressed as: \[ \text{Total amount Jane has} = 3x + 2 \] - If Jane buys 10 pairs of socks, she has $7 left over. This can be expressed as: \[ \text{Total amount Jane has} = 10y + 7 \] 2. **Set the two expressions for the total amount equal to each other:** \[ 3x + 2 = 10y + 7 \] 3. **Rearranging the equation gives us:** \[ 3x - 10y = 5 \] 4. **Now we need to find integer solutions for this equation.** To do this, we can express \(x\) in terms of \(y\): \[ 3x = 10y + 5 \implies x = \frac{10y + 5}{3} \] For \(x\) to be an integer, \(10y + 5\) must be divisible by 3. 5. **Check the divisibility condition:** - Notice that \(10y + 5 \equiv 0 \mod 3\). - Since \(10 \equiv 1 \mod 3\), we have: \[ 10y + 5 \equiv y + 2 \equiv 0 \mod 3 \implies y \equiv 1 \mod 3 \] This means \(y\) can take values like \(1, 4, 7, 10, \ldots\). 6. **Now, substitute \(y\) back to find \(x\):** - For \(y = 1\): \[ x = \frac{10(1) + 5}{3} = \frac{15}{3} = 5 \] - For \(y = 4\): \[ x = \frac{10(4) + 5}{3} = \frac{45}{3} = 15 \] - For \(y = 7\): \[ x = \frac{10(7) + 5}{3} = \frac{75}{3} = 25 \] - And so on. 7. **Now, calculate the total amount Jane has for different values of \(y\):** - For \(y = 1\): Total amount = \(3(5) + 2 = 15 + 2 = 17\) - For \(y = 4\): Total amount = \(3(15) + 2 = 45 + 2 = 47\) - For \(y = 7\): Total amount = \(3(25) + 2 = 75 + 2 = 77\) 8. **Check which of these amounts satisfies the condition for socks:** - For \(y = 1\): \(10(1) + 7 = 10 + 7 = 17\) (valid) - For \(y = 4\): \(10(4) + 7 = 40 + 7 = 47\) (valid) - For \(y = 7\): \(10(7) + 7 = 70 + 7 = 77\) (valid) 9. **Conclusion:** The possible amounts Jane could have are 17, 47, and 77 dollars. Among the options given, the only valid amount that fits the conditions is 47 dollars. **Final Answer: 47 dollars.**
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