Shortcut Trick
If the entire ₹ 600 was lent at 5%, the interest would be 5% of 600 = ₹ 30.
Actual interest received is ₹ 62, so extra interest = 62 − 30 = ₹ 32.
This extra ₹ 32 is due to the additional 8% (13% − 5%) rate on the second part.
8% of the second part = 32 ⇒ Second part = (32 × 100) ÷ 8 = ₹ 400.
First part = 600 − 400 = ₹ 200.
∴ The correct answer is ₹ 200 and ₹ 400.
Alternate Method
Given:
Total sum (P) = ₹ 600
Rates of interest = 5% and 13%
Total simple interest = ₹ 62
Time (T) = 1 year
Formula Used:
Simple Interest (I) = (P × R × T) ÷ 100
Interest at 5%
₹30
Interest at 13%
₹78
Actual Interest
₹62
78 − 62
16
62 − 30
32
Investment Ratio = 1 : 2
Calculations:
Using the Alligation method on the interest values instead of percentages to avoid fractions:
⇒ If the entire sum was invested at 5%, interest = (600 × 5) ÷ 100 = ₹ 30
⇒ If the entire sum was invested at 13%, interest = (600 × 13) ÷ 100 = ₹ 78
⇒ The actual interest earned is ₹ 62.
⇒ Applying alligation cross-difference, ratio of investments = (78 − 62) : (62 − 30) = 16 : 32 = 1 : 2
⇒ Total ratio parts = 1 + 2 = 3 parts
⇒ 3 parts = ₹ 600 ⇒ 1 part = ₹ 200
⇒ Sum lent at 5% (1 part) = ₹ 200
⇒ Sum lent at 13% (2 parts) = 200 × 2 = ₹ 400
∴ The correct answer is ₹ 200 and ₹ 400.
Additional Information
Simple Interest (SI)
It is calculated only on the principal amount. Formula: I = (P × R × T) ÷ 100, where P = Principal, R = Rate of Interest, T = Time period in years.
Rule of Alligation
When two ingredients or investments are mixed, the ratio of their quantities is inversely proportional to the differences in their unit prices or percentages. Quantity Ratio = (Dearer − Mean) : (Mean − Cheaper).
Assumption (Shift) Method
This method assumes the entire quantity belongs to one category to find the net deviation. It is often much faster than framing equations and helps avoid complex fractions in mixture and interest problems.