Let's solve the problem step-by-step:
### Part (a)
We need to find the rate of interest per annum when Rs 168 is paid as interest on Rs 700 for 16 months.
#### Step 1: Convert time from months to years
Time (in months) = 16 months
Since 1 year = 12 months,
[tex]\[
\text{Time (in years)} = \frac{16}{12} = \frac{4}{3} \text{ years}
\][/tex]
#### Step 2: Use the formula for simple interest
We use the formula for simple interest given by:
[tex]\[
I = P \cdot R \cdot T
\][/tex]
where:
- [tex]\( I \)[/tex] is the interest,
- [tex]\( P \)[/tex] is the principal,
- [tex]\( R \)[/tex] is the rate of interest per annum,
- [tex]\( T \)[/tex] is the time in years.
Rearranging the formula to solve for [tex]\( R \)[/tex]:
[tex]\[
R = \frac{I}{P \cdot T} \times 100
\][/tex]
#### Step 3: Substitute the values into the formula
Given:
- Interest, [tex]\( I = 168 \)[/tex] Rs
- Principal, [tex]\( P = 700 \)[/tex] Rs
- Time, [tex]\( T = \frac{4}{3} \)[/tex] years
Substitute these values into the formula:
[tex]\[
R = \frac{168}{700 \times \frac{4}{3}} \times 100
\][/tex]
#### Step 4: Perform the calculation
First, calculate the denominator:
[tex]\[
700 \times \frac{4}{3} = \frac{2800}{3}
\][/tex]
Now substitute:
[tex]\[
R = \frac{168}{\frac{2800}{3}} \times 100 = \frac{168 \times 3}{2800} \times 100 = \frac{504}{2800} \times 100 = \frac{18}{100} \times 100 = 18
\][/tex]
Thus, the rate of interest per annum is [tex]\( 18\% \)[/tex].
### Part (b)
We need to find the rate of interest per annum when Rs 78 is paid as interest on Rs 400 for [tex]\(1 \frac{1}{2}\)[/tex] years.
#### Step 1: Convert mixed fraction to improper fraction
[tex]\[
1 \frac{1}{2} \text{ years} = 1 + \frac{1}{2} = \frac{3}{2} \text{ years}
\][/tex]
#### Step 2: Use the formula for simple interest
Using the same formula for simple interest:
[tex]\[
I = P \cdot R \cdot T
\][/tex]
Rearranging to solve for [tex]\( R \)[/tex]:
[tex]\[
R = \frac{I}{P \cdot T} \times 100
\][/tex]
#### Step 3: Substitute the values into the formula
Given:
- Interest, [tex]\( I = 78 \)[/tex] Rs
- Principal, [tex]\( P = 400 \)[/tex] Rs
- Time, [tex]\( T = \frac{3}{2} \)[/tex] years
Substitute these values into the formula:
[tex]\[
R = \frac{78}{400 \times \frac{3}{2}} \times 100
\][/tex]
#### Step 4: Perform the calculation
First, calculate the denominator:
[tex]\[
400 \times \frac{3}{2} = 600
\][/tex]
Now substitute:
[tex]\[
R = \frac{78}{600} \times 100 = \frac{78}{6} = 13
\][/tex]
Thus, the rate of interest per annum is [tex]\( 13\% \)[/tex].
### Summary
- The rate of interest for part (a) is [tex]\( 18\% \)[/tex] per annum.
- The rate of interest for part (b) is [tex]\( 13\% \)[/tex] per annum.
