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Mathematics

Write true (T) or false (F) :

(i) If a, b, c, d are in proportion, then ac = bd.

(ii) If a : b : : c : d, then a, b, c, d are said to be in absolute proportion.

(iii) If a, b, c, are in continued proportion, then the mean proportion b = a+c2\dfrac{a + c}{2}.

(iv) If x is the third proportional to a, b, then a : b : : b : x.

(v) 1, 2, 3, 4, are in proportion.

Ratio Proportion

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Answer

(i) False
Reason — For a, b, c, d to be in proportion (a : b :: c : d), the rule is Product of Extremes = Product of Means. This means a x d = b x c, or ad = bc. The statement says ac = bd, which is incorrect.

(ii) False
Reason — When four terms are in the form a : b :: c : d, they are simply said to be in proportion. There is no standard mathematical term called "absolute proportion" used in this context.

(iii) False
Reason — If a, b, c are in continued proportion, then a : b :: b : c. This means b2 = ac or b=acb = \sqrt{ac}. The formula a+c2\dfrac{a+c}{2} is for the arithmetic mean, not the mean proportional.

(iv) True
Reason — By definition, if x is the third proportional to a and b, then a, b, and x are in continued proportion. In this sequence, b is the mean (repeated) term.

(v) False
Reason — To check if 1, 2, 3, 4 are in proportion, we test if 1 x 4 = 2 x 3:

Product of Extremes (1 x 4) = 4

Product of Means (2 x 3) = 6

Since 4 ≠ 6, they are not in proportion.

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