Francesca is painting a picture of the purple flowers in her garden. She mixed paint to make the perfect color for each type of flower. For the daisies, she mixed 2 ounces of white... Francesca is painting a picture of the purple flowers in her garden. She mixed paint to make the perfect color for each type of flower. For the daisies, she mixed 2 ounces of white paint and 5 ounces of purple paint. For the tulips, she mixed 3 ounces of white paint and 6 ounces of purple paint. Which flower is a darker shade of purple?
Understand the Problem
The question is asking us to determine which flower, daisies or tulips, has a darker shade of purple based on the ratios of white to purple paint used. We need to compare the ratios of purple paint to the total paint mixture for each flower to determine which has a higher concentration of purple.
Answer
The tulips have a darker shade of purple.
Answer for screen readers
The tulips have a darker shade of purple.
Steps to Solve
- Calculate the total parts for the daisy mixture.
The daisy mixture consists of 2 parts white and 3 parts purple. Total parts = white parts + purple parts Total parts = $2 + 3 = 5$
- Calculate the fraction of purple paint in the daisy mixture.
The fraction of purple paint is the number of purple parts divided by the total parts. Purple fraction (daisy) = $\frac{3}{5}$
- Calculate the total parts for the tulip mixture.
The tulip mixture consists of 3 parts white and 5 parts purple. Total parts = white parts + purple parts Total parts = $3 + 5 = 8$
- Calculate the fraction of purple paint in the tulip mixture.
The fraction of purple paint is the number of purple parts divided by the total parts. Purple fraction (tulip) = $\frac{5}{8}$
- Compare the fractions to determine which is greater.
We need to compare $\frac{3}{5}$ and $\frac{5}{8}$. To do this, we can find a common denominator. The least common multiple of 5 and 8 is 40.
Convert $\frac{3}{5}$ to a fraction with a denominator of 40: $\frac{3}{5} = \frac{3 \times 8}{5 \times 8} = \frac{24}{40}$
Convert $\frac{5}{8}$ to a fraction with a denominator of 40: $\frac{5}{8} = \frac{5 \times 5}{8 \times 5} = \frac{25}{40}$
Since $\frac{25}{40} > \frac{24}{40}$, the tulip mixture has a higher fraction of purple paint.
- Determine which flower has the darker shade of purple.
Since the tulip mixture has a higher fraction of purple paint, the tulips have a darker shade of purple.
The tulips have a darker shade of purple.
More Information
The problem involves comparing ratios, which can be done by converting them to fractions and finding a common denominator. This allows for direct comparison of the amounts of purple paint in each mixture.
Tips
A common mistake is to compare the ratios directly without converting them to fractions of a whole. For example, simply noting that 5 > 3 (comparing the purple parts) without considering the total mixture would lead to an incorrect conclusion.
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