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26 Apr 2026, 14:19
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On Sunday, a farmer brought two types of fruit to the market: apples and pears. During the day, the farmer sold \(\frac{1}{7}\) of the apples he had brought. If the farmer sold at least 1 apple and at least 1 pear that day, did the farmer bring more than 105 pieces of fruit to the market?
(1) The number of apples the farmer sold that day was \(\frac{15}{16}\) as many as the number of apples he sold on Saturday.
(2) During the day, the number of apples the farmer sold was \(\frac{6}{7}\) as many as the number of pears he sold.
(1) The number of apples the farmer sold that day was \(\frac{15}{16}\) as many as the number of apples he sold on Saturday.
(2) During the day, the number of apples the farmer sold was \(\frac{6}{7}\) as many as the number of pears he sold.
26 Apr 2026, 14:19
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Official Solution:
On Sunday, a farmer brought two types of fruit to the market: apples and pears. During the day, the farmer sold \(\frac{1}{7}\) of the apples he had brought. If the farmer sold at least 1 apple and at least 1 pear that day, did the farmer bring more than 105 pieces of fruit to the market?
Firstly, note that on Sunday, the farmer sold \(\frac{1}{7}\) of the apples he had brought, so apples brought on Sunday = \(7 *\) apples sold on Sunday.
(1) The number of apples the farmer sold that day was \(\frac{15}{16}\) as many as the number of apples he sold on Saturday.
This implies that:
apples sold on Sunday = \(\frac{15}{16} *\) apples sold on Saturday
\(\frac{\text{apples sold on Sunday}}{\text{apples sold on Saturday}} = \frac{15}{16}\)
Hence, apples sold on Sunday must be a multiple of 15.
Since apples brought on Sunday = \(7 *\) apples sold on Sunday, apples brought on Sunday must be a multiple of \(7 * 15 = 105\). Therefore, the total number of fruit he brought, apples + pears, must be at least \(105 + 1 = 106\), since he sold at least 1 pear, so he must have brought at least 1 pear. Sufficient.
(2) During the day, the number of apples the farmer sold was \(\frac{6}{7}\) as many as the number of pears he sold.
This implies that:
apples sold on Sunday = \(\frac{6}{7} *\) pears sold on Sunday
\(\frac{\text{apples sold on Sunday}}{\text{pears sold on Sunday}} = \frac{6}{7}\)
Hence, apples sold on Sunday must be a multiple of 6. Since apples brought on Sunday = \(7 *\) apples sold on Sunday, apples brought on Sunday must be a multiple of \(7 * 6 = 42\).
If the farmer sold 6 apples and 7 pears, then he brought \(7 * 6 = 42\) apples and could have brought 7 pears, for a total of 49 pieces of fruit, not more than 105.
However, if the farmer sold 18 apples and 21 pears, then he brought \(7 * 18 = 126\) apples and must have brought at least 21 pears, for a total of at least 147 pieces of fruit, more than 105. Not sufficient.
Answer: A
On Sunday, a farmer brought two types of fruit to the market: apples and pears. During the day, the farmer sold \(\frac{1}{7}\) of the apples he had brought. If the farmer sold at least 1 apple and at least 1 pear that day, did the farmer bring more than 105 pieces of fruit to the market?
Firstly, note that on Sunday, the farmer sold \(\frac{1}{7}\) of the apples he had brought, so apples brought on Sunday = \(7 *\) apples sold on Sunday.
(1) The number of apples the farmer sold that day was \(\frac{15}{16}\) as many as the number of apples he sold on Saturday.
This implies that:
apples sold on Sunday = \(\frac{15}{16} *\) apples sold on Saturday
\(\frac{\text{apples sold on Sunday}}{\text{apples sold on Saturday}} = \frac{15}{16}\)
Hence, apples sold on Sunday must be a multiple of 15.
Since apples brought on Sunday = \(7 *\) apples sold on Sunday, apples brought on Sunday must be a multiple of \(7 * 15 = 105\). Therefore, the total number of fruit he brought, apples + pears, must be at least \(105 + 1 = 106\), since he sold at least 1 pear, so he must have brought at least 1 pear. Sufficient.
(2) During the day, the number of apples the farmer sold was \(\frac{6}{7}\) as many as the number of pears he sold.
This implies that:
apples sold on Sunday = \(\frac{6}{7} *\) pears sold on Sunday
\(\frac{\text{apples sold on Sunday}}{\text{pears sold on Sunday}} = \frac{6}{7}\)
Hence, apples sold on Sunday must be a multiple of 6. Since apples brought on Sunday = \(7 *\) apples sold on Sunday, apples brought on Sunday must be a multiple of \(7 * 6 = 42\).
If the farmer sold 6 apples and 7 pears, then he brought \(7 * 6 = 42\) apples and could have brought 7 pears, for a total of 49 pieces of fruit, not more than 105.
However, if the farmer sold 18 apples and 21 pears, then he brought \(7 * 18 = 126\) apples and must have brought at least 21 pears, for a total of at least 147 pieces of fruit, more than 105. Not sufficient.
Answer: A
