Sure, let's solve each of the parts step-by-step.
### 1. Simple Interest and Amount to be Paid for [tex]$5000$[/tex] at [tex]$5 \%$[/tex] per annum for 2 years
First, we need to calculate the simple interest. The formula for simple interest is:
[tex]\[ \text{Simple Interest} = \frac{\text{Principal} \times \text{Rate} \times \text{Time}}{100} \][/tex]
Given:
- Principal [tex]\( P = \$5000 \)[/tex]
- Rate [tex]\( R = 5\% \)[/tex] per annum
- Time [tex]\( T = 2 \)[/tex] years
Now, substituting these values into the formula:
[tex]\[ \text{Simple Interest} = \frac{5000 \times 5 \times 2}{100} \][/tex]
[tex]\[ \text{Simple Interest} = \frac{50000}{100} \][/tex]
[tex]\[ \text{Simple Interest} = \$500 \][/tex]
Next, we need to calculate the total amount to be paid after the interest has been added to the principal. The formula for the amount [tex]\( A \)[/tex] is:
[tex]\[ \text{Amount} = \text{Principal} + \text{Simple Interest} \][/tex]
[tex]\[ \text{Amount} = 5000 + 500 \][/tex]
[tex]\[ \text{Amount} = \$5500 \][/tex]
So, the simple interest is [tex]\(\$500\)[/tex] and the total amount to be paid is [tex]\(\$5500\)[/tex].
### 2. Simple Interest and Amount to be Paid for [tex]$8000$[/tex] at [tex]$2 \frac{1}{2} \%$[/tex] per annum for 3 years
First, convert the mixed fraction to a decimal for the rate:
[tex]\[ 2 \frac{1}{2}\% = 2.5\% \][/tex]
Given:
- Principal [tex]\( P = \$8000 \)[/tex]
- Rate [tex]\( R = 2.5\% \)[/tex]
- Time [tex]\( T = 3 \)[/tex] years
Now, substituting these values into the formula for simple interest:
[tex]\[ \text{Simple Interest} = \frac{8000 \times 2.5 \times 3}{100} \][/tex]
[tex]\[ \text{Simple Interest} = \frac{60000}{100} \][/tex]
[tex]\[ \text{Simple Interest} = \$600 \][/tex]
Next, calculate the total amount to be paid:
[tex]\[ \text{Amount} = 8000 + 600 \][/tex]
[tex]\[ \text{Amount} = \$8600 \][/tex]
So, the simple interest is [tex]\(\$600\)[/tex] and the total amount to be paid is [tex]\(\$8600\)[/tex].
### 3. Simple Interest and Amount to be Paid for [tex]$10000$[/tex] at [tex]$2 \%$[/tex] per annum for [tex]$4 \frac{1}{2}$[/tex] years
First, convert the mixed fraction to a decimal for the time:
[tex]\[ 4 \frac{1}{2} \text{ years} = 4.5 \text{ years} \][/tex]
Given:
- Principal [tex]\( P = \$10000 \)[/tex]
- Rate [tex]\( R = 2\% \)[/tex]
- Time [tex]\( T = 4.5 \)[/tex] years
Now, substituting these values into the formula for simple interest:
[tex]\[ \text{Simple Interest} = \frac{10000 \times 2 \times 4.5}{100} \][/tex]
[tex]\[ \text{Simple Interest} = \frac{90000}{100} \][/tex]
[tex]\[ \text{Simple Interest} = \$900 \][/tex]
Next, calculate the total amount to be paid:
[tex]\[ \text{Amount} = 10000 + 900 \][/tex]
[tex]\[ \text{Amount} = \$10900 \][/tex]
So, the simple interest is [tex]\(\$900\)[/tex] and the total amount to be paid is [tex]\(\$10900\)[/tex].
### Summary:
1. For the loan of [tex]$\$[/tex]5000[tex]$ at $[/tex]5\%[tex]$ per annum for $[/tex]2[tex]$ years:
- Simple Interest: $[/tex]\[tex]$500$[/tex]
- Amount to be paid: [tex]$\$[/tex]5500[tex]$
2. For the loan of $[/tex]\[tex]$8000$[/tex] at [tex]$2.5\%$[/tex] per annum for [tex]$3$[/tex] years:
- Simple Interest: [tex]$\$[/tex]600[tex]$
- Amount to be paid: $[/tex]\[tex]$8600$[/tex]
3. For the loan of [tex]$\$[/tex]10000[tex]$ at $[/tex]2\%[tex]$ per annum for $[/tex]4.5[tex]$ years:
- Simple Interest: $[/tex]\[tex]$900$[/tex]
- Amount to be paid: [tex]$\$[/tex]10900$
