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Of the 100 flowers in a garden, \(\frac{3}{4}\) are either roses or tulips and \(\frac{3}{25}\) are either lilies or tulips. If there is at least one rose, at least one tulip, and at least one lily in the garden, how many of the flowers are tulips?
(1) The ratio of the number of roses to the number of tulips is less than 13 to 2.
(2) \(\frac{13}{20}\) of the flowers in the garden are either roses or lilies.
(1) The ratio of the number of roses to the number of tulips is less than 13 to 2.
(2) \(\frac{13}{20}\) of the flowers in the garden are either roses or lilies.
29 Jul 2024, 06:08
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Official Solution:
Of the 100 flowers in a garden, \(\frac{3}{4}\) are either roses or tulips and \(\frac{3}{25}\) are either lilies or tulips. If there is at least one rose, at least one tulip, and at least one lily in the garden, how many of the flowers are tulips?
• \(\frac{3}{4}\) are either roses or tulips, which implies \(R + T = 75\).
• \(\frac{3}{25}\) are either lilies or tulips, which implies \(L + T = 12\).
Observe that the above indicates that there must be some other kinds of flowers in the garden, for example, orchids.
(1) The ratio of the number of roses to the number of tulips is less than 13 to 2.
This implies that \(\frac{R}{T} < \frac{13}{2}\), which gives \(2R < 13T\).
Substituting \(R = 75 - T\), we get:
\(2(75 - T) < 13T\)
\(150 - 2T < 13T\)
\(150 < 15T\)
\(T > 10\)
Since there is at least one lily, from \(L + T = 12\) we get \(T = 11\). This is sufficient.
(2) \(\frac{13}{20}\) of the flowers in the garden are either roses or lilies.
This implies that \(R + L = 65\). Subtracting this from \(R + T = 75\), we get \(T - L = 10\). Adding this to \(L + T = 12\), we get \(2T = 22\), which gives \(T = 11\). This is sufficient too.
Answer: D
Of the 100 flowers in a garden, \(\frac{3}{4}\) are either roses or tulips and \(\frac{3}{25}\) are either lilies or tulips. If there is at least one rose, at least one tulip, and at least one lily in the garden, how many of the flowers are tulips?
• \(\frac{3}{4}\) are either roses or tulips, which implies \(R + T = 75\).
• \(\frac{3}{25}\) are either lilies or tulips, which implies \(L + T = 12\).
Observe that the above indicates that there must be some other kinds of flowers in the garden, for example, orchids.
(1) The ratio of the number of roses to the number of tulips is less than 13 to 2.
This implies that \(\frac{R}{T} < \frac{13}{2}\), which gives \(2R < 13T\).
Substituting \(R = 75 - T\), we get:
\(2(75 - T) < 13T\)
\(150 - 2T < 13T\)
\(150 < 15T\)
\(T > 10\)
Since there is at least one lily, from \(L + T = 12\) we get \(T = 11\). This is sufficient.
(2) \(\frac{13}{20}\) of the flowers in the garden are either roses or lilies.
This implies that \(R + L = 65\). Subtracting this from \(R + T = 75\), we get \(T - L = 10\). Adding this to \(L + T = 12\), we get \(2T = 22\), which gives \(T = 11\). This is sufficient too.
Answer: D
