Question

question 6 (multiple choice worth 5 points) the dishes have been sorted into cups and plates. the number of plates is four less than two times the number of cups. the dishes are 60% plates. how many cups are there? 3 6 8 9

Explanation:

Step1: Define variables

Let \( c \) be the number of cups and \( p \) be the number of plates.
From the problem, we have two equations:

1. \( p = 2c - 4 \) (The number of plates is four less than two times the number of cups)
2. \( \frac{p}{p + c}= 0.6 \) (The dishes are 60% plates, so the ratio of plates to total dishes is 0.6)

Step2: Substitute \( p \) from first equation into second

Substitute \( p = 2c - 4 \) into \( \frac{p}{p + c}= 0.6 \):

$$ \frac{2c - 4}{(2c - 4)+ c}= 0.6 $$

Simplify the denominator: \( (2c - 4)+ c=3c - 4 \)
So the equation becomes:

$$ \frac{2c - 4}{3c - 4}= 0.6 $$

Step3: Solve for \( c \)

Multiply both sides by \( 3c - 4 \):

$$ 2c - 4 = 0.6(3c - 4) $$

Expand the right - hand side: \( 2c - 4=1.8c - 2.4 \)
Subtract \( 1.8c \) from both sides: \( 2c-1.8c - 4=1.8c - 1.8c- 2.4 \)
\( 0.2c - 4=- 2.4 \)
Add 4 to both sides: \( 0.2c-4 + 4=-2.4 + 4 \)
\( 0.2c = 1.6 \)
Divide both sides by 0.2: \( c=\frac{1.6}{0.2}=8 \)

Answer:

The number of cups is 8. So the correct option (assuming the options are related to the number of cups and one of them is 8) is the one with 8 (e.g., if the options are A. 3, B. 6, C. 8, D. 9, then the answer is C. 8).