If `a:b=4:5`, then what will be the ratio of `(a/b):(b/a)`?

Options

A. `4:5`
B. `5:4`
C. `16:25`
D. `1:1`

Answer

To solve the problem step-by-step, we will start with the given ratio and then find the required ratio.

### Step 1: Understand the given ratio
We are given that \( a:b = 4:5 \). This means that:
\[ \frac{a}{b} = \frac{4}{5} \]

### Step 2: Find the reciprocal of the ratio
The reciprocal of the ratio \( a:b \) is \( b:a \). Therefore:
\[ \frac{b}{a} = \frac{5}{4} \]

### Step 3: Set up the required ratio
We need to find the ratio of \( \frac{a}{b} : \frac{b}{a} \). This can be expressed as:
\[ \frac{a}{b} : \frac{b}{a} = \frac{a}{b} \div \frac{b}{a} \]

### Step 4: Simplify the expression
Using the values we found:
\[ \frac{a}{b} \div \frac{b}{a} = \frac{a}{b} \times \frac{a}{b} = \frac{a^2}{b^2} \]

### Step 5: Substitute the values
Substituting the values of \( \frac{a}{b} \) and \( \frac{b}{a} \):
\[ \frac{4/5}{5/4} = \frac{4}{5} \times \frac{4}{5} = \frac{16}{25} \]

### Step 6: Find the ratio
Now, we can express this as:
\[ \frac{a}{b} : \frac{b}{a} = \frac{4}{5} : \frac{5}{4} \]
To express this in simplest form, we can multiply both parts by \( 20 \) (the least common multiple of 5 and 4):
\[ \frac{4 \times 20}{5 \times 20} : \frac{5 \times 20}{4 \times 20} = 16 : 25 \]

### Step 7: Final answer
Thus, the ratio \( (a/b) : (b/a) \) simplifies to:
\[ 1 : 1 \]