Question

The colored roses in the bouquet of flowers are red, yellow and pink. The ratio of the number of red to the number of yellow to the number of Pink in the bouquet is 7:4:6, respectively. If there are more than 7 yellow-colored roses, what is the minimum number of total roses in the bouquet?

• A. 8
• B. 12
• C. 14
• D. 24
• E. 34

Hint:

In ratio problems with a "minimum" or "maximum" condition, first represent the quantities using a common multiplier (\(k\)). Then, use the given condition to form an inequality for \(k\). Find the smallest integer value of \(k\) that satisfies the inequality to find the minimum total.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
This is a problem about ratios and finding a minimum value that satisfies a given condition. The number of roses of each color must be an integer, and they must maintain the given ratio.
Step 2: Detailed Explanation:
Let the number of red, yellow, and pink roses be \(R\), \(Y\), and \(P\). The ratio is given as \(R:Y:P = 7:4:6\). This means that the number of roses of each color can be represented as: \[ R = 7k \] \[ Y = 4k \] \[ P = 6k \] where \(k\) must be a positive integer, as we are dealing with a number of flowers.
We are given the condition that there are "more than 7 yellow-colored roses." \[ Y>7 \] Substitute the expression for \(Y\): \[ 4k>7 \] To find the minimum integer value of \(k\) that satisfies this inequality, we divide by 4: \[ k>\frac{7}{4} \] \[ k>1.75 \] Since \(k\) must be an integer, the smallest integer value for \(k\) that is greater than 1.75 is \(k=2\).
Now we need to find the minimum number of total roses in the bouquet. The total number of roses is \(T = R + Y + P = 7k + 4k + 6k = 17k\). To find the minimum total, we use the minimum possible integer value for \(k\), which is 2. \[ T_{min} = 17 \times 2 = 34 \] Step 3: Final Answer:
The minimum number of total roses in the bouquet is 34. This corresponds to option (E).