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Arithmetic I

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Proportion & Variation
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CAT 2025 Lesson : Proportion & Variation - Componedo & Dividendo

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1.2 Basic Properties

If
aa, bb, cc and dd are in proportion such that the Base Identity is ab=cd\dfrac{a}{b} = \dfrac{c}{d}, then the following are true.

Properties Identity Operation on Base identity
Invertendo ba=dc\dfrac{b}{a} = \dfrac{d}{c} Reciprocal
Alternendo ac=bd\dfrac{a}{c} = \dfrac{b}{d} Switching numerator for denominator
Componendo a+bb=c+dd\dfrac{a + b}{b} = \dfrac{c + d}{d} Adding 11 to both sides
Dividendo abb=cdd\dfrac{a - b}{b} = \dfrac{c - d}{d} Subtracting 11 from both sides
Componendo & Dividendo a+bab=c+dcd\dfrac{a + b}{a - b} = \dfrac{c + d}{c - d} Dividing Componendo and Dividendo
Note: You need not memorise the names of these identities.

1.3 Other Properties

1.3.1 Componendo & Dividendo with different coefficients

Property: If
22 or more ratios are equal, such that ab=cd=ef=\dfrac{a}{b} = \dfrac{c}{d} = \dfrac{e}{f} = ..., and p,q,rp, q, r and ss are real numbers, then pa+qbra+sb=pc+qdrc+sd=pe+qfre+sf=\dfrac{pa + qb}{ra + sb} = \dfrac{pc + qd}{rc + sd} = \dfrac{pe + qf}{re + sf} = ...

Note the following for the above identity
1) There is no constant term added or subtracted.
2) The coefficients of variables, i.e.
p,q,rp, q, r and ss, are applied in a uniform way in all the ratios.
3) A ratio will remain unchanged if a and b are replaced with c and d respectively or e and f respectively.

For instance if,
ab=cd=ef=\dfrac{a}{b} = \dfrac{c}{d} = \dfrac{e}{f} = ..., then 4a+3b7a5b=4c+3d7c5d=4e+3f7e5f=\dfrac{4a + 3b}{7a - 5b} = \dfrac{4c + 3d}{7c - 5d} = \dfrac{4e + 3f}{7e - 5f} = ....

Note: The ratios derived using componendo and dividendo need not equal the original ratio of
ab\dfrac{a}{b}.

Example 5

If ab=cd\dfrac{a}{b} = \dfrac{c}{d}, then 5b7a4b=\dfrac{5b}{7a - 4b} = ?
(1) ab\dfrac{a}{b}           (2) ac\dfrac{a}{c}           (3) 5c7c4d\dfrac{5c}{7c - 4d}           (4) 5d7c4d\dfrac{5d}{7c - 4d}           

Solution

As
ab=cd\dfrac{a}{b} = \dfrac{c}{d}, in the ratio 5b7a4b\dfrac{5b}{7a - 4b}, the variables a and b can be replaced by c and d respectively.


5b7a4b=5d7c4d\dfrac{5b}{7a - 4b} = \dfrac{5d}{7c - 4d}

Answer: (4)
5d7c4d\dfrac{5d}{7c - 4d}


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