Question

a type of green paint is made by mixing 2 cups of yellow with 3.5 cups of blue. match each mixture with its description. (lesson 2-1)
keyboard help
1 cup of
yellow and
1.75 cups of
blue
4 cups of
yellow and 7
cups of blue
a mixture that will make the same shade of
green but a smaller amount.
a mixture that will make the same shade of
green but a larger amount.
a mixture that will make a different shade of
green that is more yellow.
a mixture that will make a different shade of
green that is bluer.

Explanation:

To solve this, we first find the ratio of yellow to blue in the original mixture. The original ratio is \( \frac{\text{yellow}}{\text{blue}}=\frac{2}{3.5}=\frac{2}{\frac{7}{2}}=\frac{4}{7}\approx0.571 \).

For "1 cup of yellow and 1.75 cups of blue":

Calculate the ratio: \( \frac{1}{1.75}=\frac{1}{\frac{7}{4}}=\frac{4}{7}\approx0.571 \). Wait, no, wait. Wait, 1.75 is \( \frac{7}{4} \), so \( \frac{1}{1.75}=\frac{4}{7}\approx0.571 \), but wait, original ratio is \( \frac{2}{3.5}=\frac{4}{7} \). Wait, no, 2 cups yellow and 3.5 cups blue: \( \frac{2}{3.5}=\frac{4}{7}\approx0.571 \). For 1 yellow and 1.75 blue: \( \frac{1}{1.75}=\frac{4}{7}\approx0.571 \), but the amount is smaller (1 + 1.75 = 2.75, original is 2 + 3.5 = 5.5, so half the amount). Wait, maybe I miscalculated. Wait 3.5 is 7/2, so 2/(7/2)=4/7. 1.75 is 7/4, so 1/(7/4)=4/7. So the ratio is the same, and the total amount is 1 + 1.75 = 2.75, original is 2 + 3.5 = 5.5, so it's half. So this should match "A mixture that will make the same shade of green but a smaller amount."

For "4 cups of yellow and 7 cups of blue":

Calculate the ratio: \( \frac{4}{7}\approx0.571 \), same as original. The total amount is 4 + 7 = 11, which is double the original (5.5 * 2 = 11). So this should match "A mixture that will make the same shade of green but a larger amount."

Wait, but maybe there are other mixtures. But based on the given ones:

1. 1 cup yellow, 1.75 blue: ratio \( \frac{1}{1.75}=\frac{4}{7} \) (same as original \( \frac{2}{3.5}=\frac{4}{7} \)), amount is smaller (2.75 vs 5.5). So matches "A mixture that will make the same shade of green but a smaller amount."

1. 4 cups yellow, 7 cups blue: ratio \( \frac{4}{7} \) (same as original), amount is larger (11 vs 5.5). So matches "A mixture that will make the same shade of green but a larger amount."

If there were a mixture with a higher yellow ratio (e.g., more yellow relative to blue), it would be more yellow. A higher blue ratio (more blue relative to yellow) would be bluer.

But for the given two:

• 1 yellow, 1.75 blue: same shade, smaller amount.

• 4 yellow, 7 blue: same shade, larger amount.

Answer:

To solve this, we first find the ratio of yellow to blue in the original mixture. The original ratio is \( \frac{\text{yellow}}{\text{blue}}=\frac{2}{3.5}=\frac{2}{\frac{7}{2}}=\frac{4}{7}\approx0.571 \).

For "1 cup of yellow and 1.75 cups of blue":

Calculate the ratio: \( \frac{1}{1.75}=\frac{1}{\frac{7}{4}}=\frac{4}{7}\approx0.571 \). Wait, no, wait. Wait, 1.75 is \( \frac{7}{4} \), so \( \frac{1}{1.75}=\frac{4}{7}\approx0.571 \), but wait, original ratio is \( \frac{2}{3.5}=\frac{4}{7} \). Wait, no, 2 cups yellow and 3.5 cups blue: \( \frac{2}{3.5}=\frac{4}{7}\approx0.571 \). For 1 yellow and 1.75 blue: \( \frac{1}{1.75}=\frac{4}{7}\approx0.571 \), but the amount is smaller (1 + 1.75 = 2.75, original is 2 + 3.5 = 5.5, so half the amount). Wait, maybe I miscalculated. Wait 3.5 is 7/2, so 2/(7/2)=4/7. 1.75 is 7/4, so 1/(7/4)=4/7. So the ratio is the same, and the total amount is 1 + 1.75 = 2.75, original is 2 + 3.5 = 5.5, so it's half. So this should match "A mixture that will make the same shade of green but a smaller amount."

For "4 cups of yellow and 7 cups of blue":

Calculate the ratio: \( \frac{4}{7}\approx0.571 \), same as original. The total amount is 4 + 7 = 11, which is double the original (5.5 * 2 = 11). So this should match "A mixture that will make the same shade of green but a larger amount."

Wait, but maybe there are other mixtures. But based on the given ones:

1. 1 cup yellow, 1.75 blue: ratio \( \frac{1}{1.75}=\frac{4}{7} \) (same as original \( \frac{2}{3.5}=\frac{4}{7} \)), amount is smaller (2.75 vs 5.5). So matches "A mixture that will make the same shade of green but a smaller amount."

1. 4 cups yellow, 7 cups blue: ratio \( \frac{4}{7} \) (same as original), amount is larger (11 vs 5.5). So matches "A mixture that will make the same shade of green but a larger amount."

If there were a mixture with a higher yellow ratio (e.g., more yellow relative to blue), it would be more yellow. A higher blue ratio (more blue relative to yellow) would be bluer.

But for the given two:

• 1 yellow, 1.75 blue: same shade, smaller amount.

• 4 yellow, 7 blue: same shade, larger amount.