If p : q = `1(1)/(3) : 1(1)/(2)` and q : r `= (1)/(2) : (1)/(3)` , find p : r

Answer

To solve the problem of finding the ratio \( p : r \) given the ratios \( p : q = 1\frac{1}{3} : 1\frac{1}{2} \) and \( q : r = \frac{1}{2} : \frac{1}{3} \), we can follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
First, we need to convert the mixed numbers in the ratio \( p : q \) into improper fractions.

- For \( 1\frac{1}{3} \):
\[ 1\frac{1}{3} = \frac{3 \times 1 + 1}{3} = \frac{4}{3} \]

- For \( 1\frac{1}{2} \):
\[ 1\frac{1}{2} = \frac{2 \times 1 + 1}{2} = \frac{3}{2} \]

So, we have:
\[ p : q = \frac{4}{3} : \frac{3}{2} \]

### Step 2: Write the Ratio in Fraction Form
We can express the ratio \( p : q \) as:
\[ \frac{p}{q} = \frac{4/3}{3/2} \]

### Step 3: Simplify the Fraction
To simplify \( \frac{4/3}{3/2} \), we multiply by the reciprocal of the denominator:
\[ \frac{p}{q} = \frac{4}{3} \times \frac{2}{3} = \frac{8}{9} \]

### Step 4: Write the Ratio \( q : r \) in Fraction Form
Next, we express the ratio \( q : r = \frac{1}{2} : \frac{1}{3} \) in fraction form:
\[ \frac{q}{r} = \frac{1/2}{1/3} \]

### Step 5: Simplify the Fraction
To simplify \( \frac{1/2}{1/3} \), we multiply by the reciprocal of the denominator:
\[ \frac{q}{r} = \frac{1}{2} \times \frac{3}{1} = \frac{3}{2} \]

### Step 6: Find \( p : r \)
Now we have:
\[ \frac{p}{q} = \frac{8}{9} \quad \text{and} \quad \frac{q}{r} = \frac{3}{2} \]

To find \( \frac{p}{r} \), we can use the relationship:
\[ \frac{p}{r} = \frac{p}{q} \times \frac{q}{r} \]

Substituting the values we found:
\[ \frac{p}{r} = \frac{8}{9} \times \frac{3}{2} \]

### Step 7: Simplify the Result
Now, we multiply:
\[ \frac{p}{r} = \frac{8 \times 3}{9 \times 2} = \frac{24}{18} \]

### Step 8: Reduce the Fraction
We can reduce \( \frac{24}{18} \) by dividing both the numerator and denominator by their greatest common divisor (GCD), which is 6:
\[ \frac{p}{r} = \frac{24 \div 6}{18 \div 6} = \frac{4}{3} \]

### Final Result
Thus, the ratio \( p : r \) is:
\[ p : r = 4 : 3 \]