During a vacation together, Bob spent twice as much as Josh, who spent four times as much as Ralph. If Bob and Ralph together spent $180, how much did Josh spend?

Options

A. 20
B. 80
C. 120
D. 160

Answer

To solve the problem step by step, we will define the variables and set up equations based on the relationships given in the question.

### Step 1: Define the Variables
Let:
- \( B \) = amount spent by Bob
- \( J \) = amount spent by Josh
- \( R \) = amount spent by Ralph

### Step 2: Set Up the Equations
According to the problem:
1. Bob spent twice as much as Josh:
\[ B = 2J \quad \text{(Equation 1)} \]
2. Josh spent four times as much as Ralph:
\[ J = 4R \quad \text{(Equation 2)} \]
3. Bob and Ralph together spent $180:
\[ B + R = 180 \quad \text{(Equation 3)} \]

### Step 3: Substitute Equation 1 and Equation 2 into Equation 3
From Equation 1, we know:
\[ B = 2J \]
From Equation 2, we can express \( R \) in terms of \( J \):
\[ R = \frac{J}{4} \]

Now substitute \( B \) and \( R \) into Equation 3:
\[ 2J + \frac{J}{4} = 180 \]

### Step 4: Solve the Equation
To solve for \( J \), first, we need a common denominator for the left side. The common denominator for 1 and 4 is 4:
\[ 2J = \frac{8J}{4} \]
So we rewrite the equation:
\[ \frac{8J}{4} + \frac{J}{4} = 180 \]
Combine the fractions:
\[ \frac{9J}{4} = 180 \]

Now, multiply both sides by 4 to eliminate the fraction:
\[ 9J = 720 \]

Now, divide both sides by 9:
\[ J = \frac{720}{9} = 80 \]

### Conclusion
Josh spent \( J = 80 \).