Math Step by step answer: Some landscapers just finished planting a new flower garden. They used pink, yellow, and blue flowers. For every 6 pink flowers, There are 5 yellow flowers and 7 blue flowers. There are 450 flowers altogether. How many of the flowers are yellow?
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To determine the number of yellow flowers, we need to follow these steps:
Step 1: Understand the problem.
The problem states that there are three types of flowers: pink, yellow, and blue. The ratio for pink, yellow, and blue is given as 6:5:7. We are also told that there are a total of 450 flowers.
Step 2: Assign variables.
Let's assign variables to represent the number of pink, yellow, and blue flowers. Let P represent the number of pink flowers, Y for yellow flowers, and B for blue flowers.
Step 3: Set up the equations.
Based on the given ratio, we have:
P : Y : B = 6 : 5 : 7
Step 4: Calculate the ratios.
To find the total number of flowers, add the three quantities together:
P + Y + B = 450
Step 5: Use the equations to solve for the unknown.
Using the ratio, we know that for every 6 pink flowers, there are 5 yellow flowers. This can be expressed as: P/Y = 6/5, or simply P = (6/5)Y.
Similarly, for every 6 pink flowers, there are 7 blue flowers. This can be expressed as: P/B = 6/7, or simply P = (6/7)B.
Step 6: Solve the equations.
Let's substitute the values of P from the ratio equations into the total number of flowers equation:
(6/5)Y + Y + (6/7)B = 450
To simplify the equation, we can multiply through by the common denominators:
(42/35)Y + 35/35Y + (30/35)B = 450
(42Y + 35Y + 30B)/35 = 450
Combining like terms:
(77Y + 30B)/35 = 450
Multiplying both sides of the equation by 35:
77Y + 30B = 15750
Step 7: Continue solving the equations.
Since we don't have enough information to fully solve the equation, we'll use trial and error using multiples of 5 for Y and multiples of 7 for B that also satisfy the equation.
Since we know that there are 450 flowers in total, we can use this information to reduce the number of possibilities to try.
Let's start with Y = 5:
77(5) + 30B = 15750
385 + 30B = 15750
30B = 15750 - 385
30B = 15365
The result is not a multiple of 30, so we move on to the next possibility.
Let's try Y = 10:
77(10) + 30B = 15750
770 + 30B = 15750
30B = 15750 - 770
30B = 14980
The result is not a multiple of 30, so we continue trying other values of Y.
Finally, we find that Y = 15:
77(15) + 30B = 15750
1155 + 30B = 15750
30B = 15750 - 1155
30B = 14595
This result is a multiple of 30, so the solution is valid.
Step 8: Determine the answer.
Since we found that Y = 15 and this satisfies the equation, the number of yellow flowers is 15.
Step 1: Find the total number of flowers for each color.
Pink: 6 x 75 = 450
Yellow: 5 x 75 = 375
Blue: 7 x 75 = 525
Step 2: Subtract the total number of pink and blue flowers from the total number of flowers.
450 - (450 + 525) = 375
Step 3: The answer is 375 yellow flowers.