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Formulae

1. Ratio:
The ratio of two quantities a and b in the same units, is the fraction $$a \over b$$and we write it as a : b.
In the ratio a : b, we call a as the first term or antecedent and b, the second term or consequent.
Eg. The ratio 5 : 9 represents $$5 \over 9$$ with antecedent = 5, consequent = 9.
Rule: The multiplication or division of each term of a ratio by the same non-zero number does not affect the ratio.
Eg. 4 : 5 = 8 : 10 = 12 : 15. Also, 4 : 6 = 2 : 3.

2. Proportion:
The equality of two ratios is called proportion.
If a : b = c : d, we write a : b :: c : d and we say that a, b, c, d are in proportion.
Here a and d are called extremes, while b and c are called mean terms.
Product of means = Product of extremes.
Thus, a : b :: c : d ⇔ (b x c) = (a x d).

3. Fourth Proportional:
If a : b = c : d, then d is called the fourth proportional to a, b, c.

Third Proportional:
a : b = c : d, then c is called the third proportion to a and b.

Mean Proportional:
Mean proportional between a and b is $$\sqrt {ab}$$.

4. Comparison of Ratios:
We say that (a : b) > (c : d) ⇔ $$a \over b$$ > $$c \over d$$

Compounded Ratio:
The compounded ratio of the ratios: (a : b), (c : d), (e : f) is (ace : bdf).

5. Duplicate Ratios:
Duplicate ratio of (a : b) is (a2 : b2).
Sub-duplicate ratio of (a : b) is (√a : √b).
Triplicate ratio of (a : b) is (a3 : b3).
Sub-triplicate ratio of (a : b) is (a1/3 : b1/3).
If $$a \over b$$ = $$c \over d$$ , then $${a + b} \over {a - b}$$ = $${c + d} \over {c - d}$$. [componendo and dividendo]

6. Variations:
We say that x is directly proportional to y, if x = ky for some constant k and we write, x ∝ y.
We say that x is inversely proportional to y, if xy = k for some constant k and
we write, x ∝ $$1 \over y$$

Exercise : Ratio and Proportion MCQ Questions and Answers

1. If an amount of Rs 16871308 is distributed equally amongst 47 persons, how much amount would each person get ?

 A.   Rs. 348964 B.   Rs. 358964 C.   Rs. 338964 D.   Rs. 328964

Correct Answer : B. Rs. 358964

Description :
Amount received by each person= 16871308/47 = 358964

2. A gardener wants to plant trees in his garden in rows in such away that the number of trees in each row to be the same. If there are 24 rows the number of trees in each row is 42 if there are 12 more

 A.   28 B.   63 C.   48 D.   32

Correct Answer : A. 28

Description :
Required number of trees
= 24/36 * 42 = 28.

3. Find the fourth proportional to 2.4, 4.6 and 7.6?

 A.   14.56 B.   14 C.   15.56 D.   14.657

Correct Answer : A. 14.56

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4. Raj is 16 years old and Arti is 24 years. Find the ratio of Arti's age to Raj's age.

 A.   2:3 B.   3:4 C.   3:2 D.   1:2

Correct Answer : C. 3:2

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5. Karan starts business by investing Rs. 60,000. 6 months later Shirish joins by investing Rs. 1,00,000. At end of year profit is Rs. 1,51,800. What is the Shirish's share in the profit ?

 A.   Rs. 65000 B.   Rs. 72000 C.   Rs. 60000 D.   Rs. 69000

Correct Answer : D. Rs. 69000

Description :
Ratio of the profit = Ratio of the equivalent capital of Karan and Shirish for 1 month
= 60000 x 12 : 100000 x 6
= 720000 : 600000
= 6 : 5
∴ Shirish's share in the profit = Rs. (5/11 x 151800)= Rs. 69000

6. Seats for Mathematics, Physics and Biology in a school are in the ratio 5:7:8. There is a proposal to increase these seats by 40%, 50% and 75% respectively. What will be the ratio of increased seats ?

 A.   4:5:6 B.   3:4:5 C.   1:2:3 D.   2:3:4

Correct Answer : D. 2:3:4

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7. If 12 men can reap 120 acres of land in 36 days, how many acres of land can 54 men reap in 54 days?

 A.   810 acres B.   720 acres C.   650 acres D.   890 acres

Correct Answer : A. 810 acres

Description :
12 men 120 acres 36 days
54 men ? 54 days
120 * 54/12 * 54/36
10 * 54 * 3/2
54 * 15 = 810

8. Akash, Ramesh and Kamlesh invest in a business in the ratio of 2 : 3 : 5. If the amount invested by Akash is Rs 12000, what is the total amount invested ?

 A.   Rs. 58,000 B.   Rs. 62,000 C.   Rs. 64,000 D.   Rs. 60,000

Correct Answer : D. Rs. 60,000

Description :
Total of the ratio = 2 + 3 + 5 = 10
∴ Total amt/10 x 2 = 12000
∴ Total amount = 12000 x 10/2 = Rs 60000

9. If p, q and r are positive integers and satisfy x = (p + q -r)/r = (p - q + r)/q = (q + r - p)/p, then the value of x is?

 A.   1 B.   -1 C.   -1/2 D.   1/2

Correct Answer : A. 1

Description :
When two or more ratios are equal, each of the ratios are equal to sum of the numerators divided by the sum of the denominators, provided sum of the denominators is non-zero.
Hence, x = (p + q -r)/r = (p - q + r)/q = (q + r - p)/p
=> x = (p + q - r + p - q + r + q + r - p) / (r + q + p)
=> x = (r + q + p) / (r + q + p) = 1
p + q + r is non-zero.

10. The ratio of 1.5 m to 10 cm is:

 A.   15 : 1 B.   5 : 7 C.   6 : 10 D.   10 : 5

Correct Answer : A. 15 : 1

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