PREVIOUSLY ASKED IN:
CTET 2026
Answer
13
Explanation
According to the condition, \( \frac{4+x}{10+x} = \frac{12+x}{24+x} \). Cross-multiplying: \( (4+x)(24+x) = (10+x)(12+x) \). Expanding this: \( 96 + 28x + x^2 = 120 + 22x + x^2 \). Cancelling \( x^2 \) from both sides: \( 6x = 24 \) which gives \( x = 4 \). Finally, we need the value of \( (3x+1) = 3(4) + 1 = 13 \).
Key Points
- > If four numbers a, b, c, d are in proportion, then \( \frac{a}{b} = \frac{c}{d} \).
- > A core property of proportions is that the product of extremes equals the product of means (ad = bc).
- > In such added-value ratio problems, quadratic terms (like \( x^2 \)) usually appear on both sides and cancel out naturally.
- > For quick solving, one can iteratively check small integer values (1, 2, 3, 4) for \( x \).
- > Putting \( x=4 \) gives ratios \( \frac{8}{14} = \frac{16}{28} \) (both equal to \( \frac{4}{7} \)), verifying the answer.
- > The question asks for \( (3x+1) \), not just \( x \), which is a common trick to catch careless mistakes.
Additional Information
- >## Ratio & Proportion Facts| Type of Proportion | Property | Example | |---|---|---| | Direct Proportion | a:b = c:d | 2:3 :: 4:6 | | Mean Proportional | x = √(ab) | Mean of 4 and 9 = 6 | | Third Proportional | c = b²/a | Third of 2 and 4 = 8 | | Fourth Proportional | d = bc/a | Fourth of 2,3,4 = 6 |### Memory Tips - **Extremes and Means**: In a, b, c, d
- > 'a' and 'd' are extremes, while 'b' and 'c' are means. The equation **a × d = b × c** is always applicable.