Question

grade 7: week of 12/15, math hw - day 1
directions: show all steps of your thinking and circle final answers!
problem #1 - ratios and rates with fractions (mild)
in order to make a specific shade of green paint, a painter mixes 1\frac{1}{2} quarts of blue paint, 2 cups of green paint, and \frac{1}{2} gallon of white paint. how much of each color is needed to make 100 cups of this shade of green paint?

Explanation:

Step1: Find total paint in quarts

First, convert all volumes to the same unit (quarts). We know that 1 gallon = 4 quarts, so \(\frac{1}{2}\) gallon of white paint is \(\frac{1}{2}\times4 = 2\) quarts. The blue paint is \(1\frac{1}{2}=\frac{3}{2}\) quarts, green is 2 quarts. Total paint per batch: \(\frac{3}{2}+2 + 2=\frac{3}{2}+4=\frac{3 + 8}{2}=\frac{11}{2}\) quarts.

Step2: Convert 100 cups to quarts

Since 1 quart = 4 cups, 100 cups is \(\frac{100}{4}=25\) quarts.

Step3: Find number of batches

Number of batches \(n=\frac{25}{\frac{11}{2}}=25\times\frac{2}{11}=\frac{50}{11}\) (Wait, maybe I misread the original mix. Wait, let's re - check the problem: "mixes \(1\frac{1}{2}\) quarts of blue paint, 2 cups of green paint, and \(\frac{1}{2}\) gallon of white paint". Oh! I made a mistake earlier: green is in cups, white in gallons, blue in quarts. Let's correct:

First, convert all to cups. 1 quart = 4 cups, so \(1\frac{1}{2}\) quarts of blue: \(1.5\times4 = 6\) cups. \(\frac{1}{2}\) gallon of white: \(\frac{1}{2}\times16 = 8\) cups (since 1 gallon = 16 cups). Green is 2 cups. Total per batch: \(6 + 2+8 = 16\) cups.

Step4: Find batches for 100 cups

Number of batches \(n=\frac{100}{16}=\frac{25}{4}=6.25\)

Step5: Find amount of each color

• Blue: \(6\times6.25 = 37.5\) cups (or \(37.5\div4 = 9.375\) quarts)
• Green: \(2\times6.25 = 12.5\) cups
• White: \(8\times6.25 = 50\) cups (or \(50\div16 = 3.125\) gallons)

Wait, maybe the initial unit conversion was wrong. Let's start over with correct unit conversion:

1 quart = 4 cups, 1 gallon = 16 cups.

Blue: \(1\frac{1}{2}\) quarts=\(1.5\times4 = 6\) cups.

White: \(\frac{1}{2}\) gallon=\(\frac{1}{2}\times16 = 8\) cups.

Green: 2 cups.

Total per batch: \(6 + 2+8 = 16\) cups.

To make 100 cups, the number of batches \(k=\frac{100}{16}=\frac{25}{4}=6.25\)

Blue: \(6\times6.25 = 37.5\) cups

Green: \(2\times6.25 = 12.5\) cups

White: \(8\times6.25 = 50\) cups

If we want to present in the original units:

Blue: \(37.5\) cups \(=\frac{37.5}{4}=9.375\) quarts \(=9\frac{3}{8}\) quarts

Green: \(12.5\) cups

White: \(50\) cups \(=\frac{50}{16}=\frac{25}{8}=3.125\) gallons \(=3\frac{1}{8}\) gallons

Answer:

Blue: \(37.5\) cups (or \(9.375\) quarts), Green: \(12.5\) cups, White: \(50\) cups (or \(3.125\) gallons)