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06 Aug 2026, 08:12
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A training center issued 180 course certificates last month. Some certificates were marked Advanced, and each certificate was for a course lasting either at least 6 weeks or fewer than 6 weeks. How many of the Advanced certificates were for courses lasting at least 6 weeks?
(1) 45 certificates were marked Advanced and were for courses lasting fewer than 6 weeks. These certificates represented \(\frac{3}{4}\) of all Advanced certificates.
(2) 50 certificates were not marked Advanced and were for courses lasting at least 6 weeks. These certificates represented \(\frac{10}{13}\) of all certificates for courses lasting at least 6 weeks.
(1) 45 certificates were marked Advanced and were for courses lasting fewer than 6 weeks. These certificates represented \(\frac{3}{4}\) of all Advanced certificates.
(2) 50 certificates were not marked Advanced and were for courses lasting at least 6 weeks. These certificates represented \(\frac{10}{13}\) of all certificates for courses lasting at least 6 weeks.
06 Aug 2026, 08:12
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Official Solution:
A training center issued 180 course certificates last month. Some certificates were marked Advanced, and each certificate was for a course lasting either at least 6 weeks or fewer than 6 weeks. How many of the Advanced certificates were for courses lasting at least 6 weeks?
The question asks for:
\(\{\text{Advanced and at least 6 weeks}\}\)
(1) 45 certificates were marked Advanced and were for courses lasting fewer than 6 weeks. These certificates represented \(\frac{3}{4}\) of all Advanced certificates.
This gives \(\{\text{Advanced and fewer than 6 weeks}\} = 45\).
So:
\(\{\text{Advanced}\} = \frac{\{\text{Advanced and fewer than 6 weeks}\}}{\frac{3}{4}} = \frac{45}{\frac{3}{4}} = 60\)
Therefore:
\(\{\text{Advanced and at least 6 weeks}\} = \{\text{Advanced}\} - \{\text{Advanced and fewer than 6 weeks}\}\)
\(\{\text{Advanced and at least 6 weeks}\} = 60 - 45 = 15\)
Sufficient.
(2) 50 certificates were not marked Advanced and were for courses lasting at least 6 weeks. These certificates represented \(\frac{10}{13}\) of all certificates for courses lasting at least 6 weeks.
This gives \(\{\text{Not Advanced and at least 6 weeks}\} = 50\).
So:
\(\{\text{At least 6 weeks}\} = \frac{\{\text{Not Advanced and at least 6 weeks}\}}{\frac{10}{13}} = \frac{50}{\frac{10}{13}} = 65\)
Therefore:
\(\{\text{Advanced and at least 6 weeks}\} = \{\text{At least 6 weeks}\} - \{\text{Not Advanced and at least 6 weeks}\}\)
\(\{\text{Advanced and at least 6 weeks}\} = 65 - 50 = 15\)
Sufficient.
Answer: D
A training center issued 180 course certificates last month. Some certificates were marked Advanced, and each certificate was for a course lasting either at least 6 weeks or fewer than 6 weeks. How many of the Advanced certificates were for courses lasting at least 6 weeks?
The question asks for:
\(\{\text{Advanced and at least 6 weeks}\}\)
(1) 45 certificates were marked Advanced and were for courses lasting fewer than 6 weeks. These certificates represented \(\frac{3}{4}\) of all Advanced certificates.
This gives \(\{\text{Advanced and fewer than 6 weeks}\} = 45\).
So:
\(\{\text{Advanced}\} = \frac{\{\text{Advanced and fewer than 6 weeks}\}}{\frac{3}{4}} = \frac{45}{\frac{3}{4}} = 60\)
Therefore:
\(\{\text{Advanced and at least 6 weeks}\} = \{\text{Advanced}\} - \{\text{Advanced and fewer than 6 weeks}\}\)
\(\{\text{Advanced and at least 6 weeks}\} = 60 - 45 = 15\)
Sufficient.
(2) 50 certificates were not marked Advanced and were for courses lasting at least 6 weeks. These certificates represented \(\frac{10}{13}\) of all certificates for courses lasting at least 6 weeks.
This gives \(\{\text{Not Advanced and at least 6 weeks}\} = 50\).
So:
\(\{\text{At least 6 weeks}\} = \frac{\{\text{Not Advanced and at least 6 weeks}\}}{\frac{10}{13}} = \frac{50}{\frac{10}{13}} = 65\)
Therefore:
\(\{\text{Advanced and at least 6 weeks}\} = \{\text{At least 6 weeks}\} - \{\text{Not Advanced and at least 6 weeks}\}\)
\(\{\text{Advanced and at least 6 weeks}\} = 65 - 50 = 15\)
Sufficient.
Answer: D
