Ratio & Proportion
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Introduction
RATIO: When we say that the length of a line AB is 5 centimeters, we mean that a unit of length called 1 centimeter is contained in AB five times. If we have two lines AB and CD and their lengths are 2 and 3 centimeters respectively, we say that the length of AB is 2/3 of the length of CD.
DEFINITION:
The number of times one quantity contains another quantity of the same kind is called as the ratio of the two quantities. Clearly, the ratio of two quantities is equivalent to the fraction that one quantity is of the other. Observe carefully that the two quantities must be of the same kind. There can be a ratio between Rs.20 and Rs.30, but there can be no ratio between Rs.20 and 30 mangoes.
In other words, if the values of two quantities A and B are 4 and 6 respectively, then we say that they are in the ratio 4 : 6. Ratio is the relation which one quantity bears to another of the same kind, the comparison being made by considering what multiple, part or parts, one quantity is of the other.
Since the quotient obtained on dividing one concrete quantity by another of the same kind is an abstract number, the ratio between two concrete quantities of the same kind is an abstract number. It may be an integer or fraction. Thus the ratio between Rs.5 and Rs.7 is 5 : 7.
The number of times one quantity contains another quantity of the same kind is called as the ratio of the two quantities. Clearly, the ratio of two quantities is equivalent to the fraction that one quantity is of the other. Observe carefully that the two quantities must be of the same kind. There can be a ratio between Rs.20 and Rs.30, but there can be no ratio between Rs.20 and 30 mangoes.
In other words, if the values of two quantities A and B are 4 and 6 respectively, then we say that they are in the ratio 4 : 6. Ratio is the relation which one quantity bears to another of the same kind, the comparison being made by considering what multiple, part or parts, one quantity is of the other.
Since the quotient obtained on dividing one concrete quantity by another of the same kind is an abstract number, the ratio between two concrete quantities of the same kind is an abstract number. It may be an integer or fraction. Thus the ratio between Rs.5 and Rs.7 is 5 : 7.
REPRESENTATION: The ratio of two quantities “a” and “b” is represented as a : b and read as “a is to b”. A ratio a : b can also be expressed as a/b. So if two items are in the ratio 2 : 3, we can say that their ratio is 2/3. If two terms are in the ratio 2, it means that they are in the ratio of 2/1, that is, 2 : 1.
TERMS: In a ratio a : b, a and b are called as the terms of the ratio. The term a is called as the first term or antecedent. The term b is called as the second term or consequent.
RULES:
(i) Ratio of any number of quantities is expressed after removing any common factors that all the terms of the ratio have. For example, if there are two quantities having values of 4 and 6, their ratio is 4 : 6, that is 2 : 3 after taking the common factor 2 between them out. Similarly, if there are three quantities 6, 8 and 18, there is a common factor between all three of them. So, dividing each of the three terms by 2, we get the ratio as 3 : 4 : 9.(ii) If two quantities whose values are A and B respectively are in the ratio a : b, since we know that some common factor k > 0 would have been removed from A and B to get the ratio a : b, we can write the original values of the two quantities as A = ak and B = bk respectively. For example, if the salaries of two persons are in the ratio 7 : 5, we can write their individual salaries as 7k and 5k respectively.(iii) A ratio is said to be a ratio of greater or lesser inequality or of equality according as antecedent is greater than, less than or equal to the consequent. In other words,
· The ratio a : b where a > b is called as ratio of greater inequality.
· The ratio a : b where a < b is called as ratio of lesser inequality.
· The ratio a : b where a = b is called as ratio of equality.
(iv) From the above rule, we can find that a ratio of greater inequality is diminished and a ratio of lesser inequality is increased by adding the same quantity to both terms, that is, in the ratio a : b, when we add the same quantity x ( positive ) to both the terms of the ratio, we have the following results· If a < b then ( a + x ) : ( b + x ) > a : b.
· If a > b then ( a + x ) : ( b + x ) < a : b.
· If a = b then ( a + x ) : ( b + x ) = a : b.
(v) The value of a ratio remains unchanged, if each one of its terms is multiplied or divided by a same non-zero number. For example, 4 : 5 = 8 : 10 = 12 : 15 etc.
COMPOUND RATIO: Ratios are compounded by multiplying together the antecedents for a new antecedent and the consequents for a new consequent. For example, the compounded ratio of the ratios ( a : b ), ( c : d ) and ( e : f ) is ( ace : bdf ).
· a2 : b2 is called as the duplicate ratio of a : b.
· a3 : b3 is called as the triplicate ratio of a : b.
· a1/2 : b1/2 is called as the sub-duplicate ratio of a : b.
· a1/3 : b1/3 is called as the sub-triplicate ratio of a : b.
INVERSE RATIO: If a : b is the given ratio, then 1/a : 1/b or b : a is called its inverse or reciprocal ratio.
PROPORTION: When two ratios are equal, then the four quantities involved in the two ratios are said to be proportional, that is, if a / b = c / d, then a, b, c and d are proportional. In other words, the equality of ratios is called as proportion.
REPRESENTATION: If the numbers a, b, c and d are said to be in proportion, then it is represented as ( a : b :: c : d ) and is read as “a is to b (is) as c is to d”. Other ways of representing the same are, ( a : b = c : d ) or ( a / b = c / d ).
TERMS: If we have a : b :: c : d, then a, b, c and d are called as terms of the proportion, where a is the first term, b is the second term, c is the third term and d is the fourth term. The first and fourth terms, that is a and d are called as the extremes or end terms of the proportion. The second and third terms that are b and c are called as the means or middle terms of the proportion. The fourth term that is d is also called as the fourth proportional.
RULES:
(a) If four quantities be in proportion, then the product of the extremes is equal to the product of the means. In general, if ( a : b :: c : d ), then ( a * d = b * c ).
(b) Three quantities of the same kind are said to be in continued proportion when the ratio of the first to the second is equal to the ratio of the second to the third. The second quantity is called as the mean proportional between the first and the third quantity. The third quantity is called as the third proportional to the first and second terms.
(c) If a : b = c : d then, b : a = d : c. This relationship is called as INVERTENDO.
(d) If a : b = c : d then, a : c = b : d. This relationship is called as ALTERNENDO.
(e) If a : b = c : d then, ( a + b ) : b = ( c + d ) : d. This relationship is called as COMPONENDO. This is obtained by adding 1 to both sides of the given relationship.
(f) If a : b = c : d then, ( a + b ) : b = ( c + d ) : d. This relationship is called as DIVIDENDO. This is obtained by subtracting 1 to both sides of the given relationship.
(g) If a : b = c : d then, ( a + b ) : ( a – b ) = ( c + d ) : ( c – d ). This relationship is called as COMPONENDO–DIVIDENDO. This is obtained by dividing the componendo and dividendo relationship.
(h) The last relationship, that is, Componendo-Dividendo is very helpful in simplifying problems. By this rule, whenever we know a / b = c / d, then we can write ( a + b ) / ( a – b ) = ( c + d ) / ( c – d ). The converse of this is also true.
(i) If a/b = c/d = e/f………, then each of these ratios is equal to (a+c+e+…)/(b+d+f+…).
VARIATION: Two quantities A and B may be such that as one quantity changes in value, the other quantity also changes in value bearing certain relationship to the change in the value of the first quantity.
DIRECT VARIATION:
(a) One quantity A is said to vary directly as another quantity B if the two quantities depend upon each other in such a manner that if B is increased in a certain ratio, A is increased in the same ratio and if B is decreased in a certain ratio, A is decreased in the same ratio.
(b) This is denoted as A # B ( A varies directly as B ).
(c) If A # B then A = k * B, where k is a constant. It is called as constant of proportionality.
(d) For example, when the quantity of sugar purchased by a housewife doubles from the normal quantity, the total amount she spends on sugar also doubles, that is, the quantity and the total amount increases ( or decreases ) in the same ratio.
(e) From the above definition of direct variation, we can see that when two quantities A and B vary directly with each other, then A/B = k or the ratio of the two quantities is a constant. Conversely, when the ratio of two quantities is a constant, we can conclude that they vary directly with each other.
(f) If X varies directly with Y and we have two sets of values of the variables X and Y, that is, X1 corresponding to Y1 and X2corresponding to Y2, then, since X # Y, we can write down
X1 X2 X1 Y1
— = — or — = —
Y1 Y2 X2 Y2
INVERSE VARIATION:
(a) One quantity A is said to vary inversely as another quantity B if the two quantities depend upon each other in such a manner that if B is increased in a certain ratio, A is decreased in the same ratio and if B is decreased in a certain ratio, A is increased in the same ratio.
(b) It is the same as saying that A varies directly with 1/B. It is denoted as, if A # 1/B, that is, A = k/B where k is constant of proportionality.
(c) For example, as the number of men doing a certain work increases, the time taken to do the work decreases and conversely, as the number of men decreases, the time taken to do the work increases.
(d) From the above definition of inverse variation, we can see that when two quantities A and B vary inversely with each other, then AB = k or the product of the two quantities is a constant. Conversely, if the product of two quantities is a constant, we can conclude that they vary inversely with each other.
(e) If X varies inversely with Y and we have two sets of values of the variables X and Y, that is, X1 corresponding to Y1 and X2corresponding to Y2, then, since X # 1/Y, we can write down
X1 Y2
— = — or X1 * Y1 = X2 * Y2
X2 Y1
JOINT VARIATION: If there are three quantities A, B and C such that A varies with B when C is constant and varies with C when B is constant, then A is said to vary jointly with B and C when both B and C are varying, that is, A # B when C is constant and A # C when B is a constant. This implies A # B * C = k * B * C where k is the constant of proportionality.
Ratio:
This is a comparison of of the sizes of two or more quantities of the same kind.
If "p" and "q" are the two quantities of the same kind as well as in the same units, the fraction p/q is called the ratio of "p" to "q".
Thus, the ratio. of "p" to "q" = p/q or p:q. The quantities "p" and "q" are called the terms of the ratio. "p" is called the first term or antecedent "q" is called the second term or consequent.
The ratio of two quantities a and b in the same units, is the fraction 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
with antecedent = 5, consequent = 9.
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.
Proportion:
This is another branch of the topic Ratio and Proportion. If two ratios are equal, then it is called proportion.
For example
Four quantities a,b,c,d are said to be in proportion if a:b=c:d.
And also it can be said as a:b :: c:d or a/b = c/d or ad=bc.
Cross product rule in Proportion
product of extremes = product of means
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).
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 ab.
Comparison of Ratios:
We say that (a : b) > (c : d)
a
>
c
.
b
d
Compounded Ratio:
The compounded ratio of the ratios: (a : b), (c : d), (e : f) is (ace : bdf).
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
=
c
, then
a + b
=
c + d
. [componendo and dividendo]
b
d
a - b
c - d
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 y
SOME POINTS TO BE REMEMBERED
1. The ratio a : b represents a fraction a/b. a is called antecedent and b is called consequent.
2. The equality of two different ratios is called proportion.
3. If a : b = c : d then a, b, c, d are in proportion. This is represented by a : b :: c : d.
4. In a : b = c : d, then we have a* d = b * c.
5. If a/b = c/d then (a + b ) / ( a – b ) = ( d + c ) / ( d – c ).
RATIO- The ratio of two quantities of the same kind is the fraction that one quantity is of the other, in other words to say, how many times a given number is in comparison to another number. A ratio between two nos. A and B is denoted by A/B
1. The two quantities must be of the same kind.
2. The units of the two quantities must be the same.
3. The ratio has no measurement.
4. The ratio remains unaltered even if both the antecedent (A) and the consequent (B) are multiplied or divided by the same no.
5 If two different ratios ( say A /B and C/D) are expressed in different units, then if we are required to combine these two ratios we will follow the following rule=
A xC / B xD The required ratio is AC / BD
6 The duplicate ratio of A/B is A2/B2 the triplicate ratio of A/B is A3/B3
7 The sub duplicate ratio of A/B is sq.root of A/ sq.root of B
8 The sub triplicate ratio of A/B is cube root of A/ cube root of B
9 To determine which of the given two ratio A/B and C/D is greater or smaller ,we compare A xD and B xC provided B>0 and D>0;
if AxC> B xD then A/B > C/D and vice versa,but if A xC= B xD then A/B = C/D
Properties of ratios.
1. Inverse ratios of two equal ratios are equal, if A/B=C/D then B/A = D/C.
2. The ratios of antecedents and consequents of two equal ratios are equal if A/B=C/D then A/C=B/D
3. If A/B=C/D THEN A+B/B=C+D/D
4. If A/B=C/D THEN A-B/B=C-D/D
5. If A/B=C/D THEN A+B/A-B=C+D/C-D
6. If A/B=C/D=E/F.....so on then each of the ratio (A/B, C/D.....etc) is equal to
sum of the numerators/sum of the denominators=A+C+E...../B+D+F......=k
PROPORTION
1 Two ratios of two terms is equal to the ratio of two other terms, then these four terms are said to be in proportion i.e. if A/B=C/D then A,B,C and D are in proportion.
A,B,C and D are called first, second, third and fourth proportional’s respectively.
A and D are called Extremes and B and C are called the Means
and it follows that A xD=B xC
2 Continued proportion: when A/B=B/C then A, B and C are said to be in continued proportion and B is called the geometric mean of A and C so it follows,
A xC=B2 ,OR square root of (A xC)=B
3 Direct proportion: if two quantities A and B are related and an increase in A decreases B and vice-versa then A and B are said to be in direct proportion. Here A is directly proportional to B is written as AµB.when a is removed equation comes to be
A = kB,where k is constant.
4. Inverse proportion: if two quantities A and B are related and an increase in A increases B and vice-versa then A and B are said to be in inverse proportion. Here A is inversely proportional to B is written as Aµ1/B or, A=k/B,where k is constant.
5 Proportional division:
It simply means a method by which a quantity may be divided into parts which bear a given ratio to one another .The parts are called proportional parts.
e.g. divide quantity "y" in the ratio a:b:c then
first part= a/(a+b+c)=y second part=b/(a+b+c)=y third part=c/(a+b+c)=y
6. If in x liters mixture of Milk and water the ratio of Milk and Water is a;b, the quantity of water to be added in order to make this ratio c: d is
X(ad-bc) / c(a+b)
7. A mixture contains milk and water in the ratio of a;b. If x liters of water is added to the mixture, milk and water become in the ratio a;c. then the quantity of milk in the mixture is given by ax / c-b and that of water is given by bx /c-b , M= x9a+b) /c-b
8. If two quantities X and Y are in the ratio x;y, then X+Y : X-Y :: x+y : x-y
9. If the sum of two numbers is A and their difference is a, then the ratio of numbers is given by A+a : A-a
EXAMPLES
1. If (x/y) = (2/3) then find the value of (3x+4y)/(4x+3y)
Sol: =(3x+4y)/(4x+3y)
Divide numerator and denominator by “y” ={3(x/y)+4y/y}/{4(x/y)+3y/y}
={3(x/y)+4}/{4(x/y)+3}
Substitute x/y= 2/3
= {3(2/3)+4}/{4(2/3)+3}
= {2+4}/{(8/3)+3}
= 6/{(8+9)/3
= 6/{17/3}
= (6x3)/17
= 18/7
2. For what value of ‘m’, will the ratio (7+m)/(12+m) be equal to 5/6?Sol: Let (7+m)/(12+m)= 5/6
6(7+m)= 5(12+m)
42+6m=60+5m
6m-5m=60-42, m=18
3.Find the value of "x" if 10:x = 5:4.
Sol: By using cross product rule, we have 5x=10 times 4
5x=40
x=40/5
x=8
4. Find the fourth proportional to 2/3, 3/7, 4,
Sol: Let the fourth proportional be "x", then 2/3, 3/7, 4, x are in proportion.
Using cross product rule, (2/3)x=(3 times 4)/7
(2/3)x=12/7
x=(12 times 3)/((7 times 2)
x= 36/14
x= 18/7
5. Find the three numbers in the ratio of 1:2:3 so that the sum of their squares is equal to 504?
Sol: let 1st no. be 1x,2 nd no. be 2x and 3rd no. be 3x
their squares- x2 , (2x)2 and (3x)2
as per the question, x2 + (2x)2+(3x)2 = 504
x2+4x2+9x2=504
14x2=504
x2=504/14=36
so, x=6
So the three no. are 1x=6,2x=12 and 3x=18
6. Find the fourth proportional to the numbers 6,8 and 15?
Sol: let K be the fourht proportional, then 6/8=15/K
Solving it we get K=(8x15)/6= 20
7. Find the mean mean proportion between 3 and 75?
Sol. this is related to continued proportion.let x be the mean proportionalx then we have
x2=3x75 or x=15
8. Divide Rs 1350 into three shares proportional to the numbers 2, 3 and 4?
Sol: 1st share= Rs 1350x(2/2+3+4)=Rs 300
2nd share = Rs1350x(3/2+3+4)=Rs 450
3rd share= Rs1350x(4/2+3+4)=Rs 600
9. A certain sum of money is divided among A,B and C such that for each rupee A has ,B has 65 paise and C has 40 paisex if C's share is Rs 8, find the sum of money?
Sol: here A:B:C = 100:65:40 = 20:13:8
now 20+13+8=41
As 8/14 of the whole sum=Rs 8
So, the whole sum=Rs 8x41/8=Rs 41
10. In 40 liters mixture of milk and water the ratio of milk and water is 3:1. how much water should be added in the mixture so that the ratio of milk to water becomes 2:1.?
Sol: here only amount of water is changing. the amount of milk remains same in both the mixtures. So, amount of milk before addition of water =(3/4)X40=30 ltrs. So amount of water is 10 ltrs.
After addition of water the ratio changes to 2:1.here the mixture has two ltrs of milk for every 1 ltr of water. Since amount of milk is 30 ltrs the amount of water has to be 15 ltr so that the ratio is 2:1. So the amount of water to be added is 15-10=5 liters.
11. A sum of Rs. 427 is to be divided among A, B and C such that 3 times A’s share, 4 tunes B’s share and 7 times C’s share are all equal. The share of C is
Sol: 3A = 4B = 7C = k,Then A = k/3, B = k/4 and C= k/7.
A : B : C = k/3 : k/4 : k/7 = 28:21 :12.
Cs share = Rs. [427 x (12/61)] = Rs. 84
12. If a+b : b+c : c+a = 6 : 7 : 8 and a + b + c = 14, then the value of c is
Sol: a/3) = (b/4) = (c/7) then a = 3k, b = 4k, c = 7k
a+b+c/c = 3k+4k+7k/7k = 14k/7k = 2
13. The least whole number which when subtracted from both the terms of the ratio 6 : 7 to give a ratio less than 16 : 21, is.
Sol: Let x is subtracted. Then, ((6 - x)/(7 - x)) < 16 / 21
21(6—x) < 16(7—x) ⇒ 5x > 14 = x > 2.8.
Least such number is 3.
14. If 15% of x is the same as 20% of y, then x : y is :
Sol: 15% of x = 2O% of y ⇒ 15x/100 = 20y/100 ⇒ x/y = 4/3
15. The ratio of income of A to that of B is 5 : 4 and the expenditure of A to that of B is 3: 2. If at the end of the year, each saves Rs, 800, the income of A is: .
Sol: Let the income of A and B be 5x and 4x and. the expenditures of A and B be 3y and 2y. Then, 5x—3y = 800 and 4x— 2y= 800.
On solving we get: x = 400. As income = 5x = Rs. 2000.
16. An alloy is to contain copper and zinc in the ratio 9:4. The zinc required (in kg) to be melted with 24 kg of copper, is 7
Sol: 9:4: 24:x ⇒ 9x = 4 * 24 ⇒ x = (4*24)/9 = 32/3 Kg. hence `0 and 1/3
17. The ratio of two numbers is 3 : 4 and their sum is 420. The greater of the two numbers is
Sol: Required number = (420 * (4/7)) = 240.
18. Rs. 730 were divided among A, B, C in such a way that if A gets Rs. 3, then B gets Rs. 4 and if B gets Rs. 3.50 then C gets Rs. 3. The share of B exceeds that of C by:
Sol: A:B = 3:4 and B:C = 7/2:3 = (8/7)*(7/2)*(8/7)*3 = 4:(24/7)
A : B : C = 3 :4: 24/7 = 21 : 28 : 24.
Bs share = Rs. [730 *(28/73)]= Rs. 280.
C’s share = Rs. [730 * (24/73)] = Rs. 240.
Difference of their shares = 40
19. If 7 : x = 17.5 : 22.5 , then the value of x is:.
Sol: 7*22.5 = x*17.5 ⇒ x = 7 * 22.5/17.5 ⇒ x = 9.
20. What number should be subtracted from both the terms of the ratio 15 : 19 so as to make it as 3 : 4 ?
Sol: Let x be subtracted. Then,
(15 - x) / (19 - x) = 3/4 ⇒ 4(15 - x) = 3(19 - x) x = 3
21. What number should be added to each of the numbers 8, 21, 13 and 31 so that the resulting numbers, in this order form a proportion?
Sol: (8+x)/(21+x) = (13+x)/(31+x)
Then, (8 + x)(31 + x) = (13 + x)(21 + x)
or39x + 248 = 34x + 273 or 5x=25 or x = 5.
22. If 0.4: 1.4: 1.4: x, the value of x is
Sol: 0.4 * x = 1.4 * 1.4 ⇒ x = (1.4*1.4)/0.4 = 4.9
23. A dog takes 3 leaps for every 5 leaps of a hare. If one leap of the dog is equal to 3 leaps of the hare, the ratio of the speed of the dog to that of the hare is:.
Sol: Dog : Hare = (3*3) leaps of hare : 5 leaps of hare = 9 : 5.
24. The salaries of A, B, and C are in the ratio of 1 : 2 : 3. The salary of B and C together is Rs. 6000. By what percent is the salary of C more than that of A?
Sol: Let the salaries of A, B, C hex, 2x and 3x respectively.
Then,2x + 3x = 6000 = x = 1200. As salary = Rs. 1200, Bs salary = Rs. 2400, and Cs salary Rs. 3600.
Excess of Cs salary over As=[(2400/1200)x100] = 200%.
25. A certain amount was divided between Salim and Rahim in the ratio of 4 : 3. If Rahim’s share was Rs. 2400, the total amount was.
Sol: Let S = 4x and R = 3x. Total amount = 7x.
Then, 3x = 2400 so x= 800.
Total amount = 7x = Rs. 5600
26. A sum of money is to the divided among F, Q andR in the ratio of 2 : 3 : 5. If the total share of P andR together is Rs 400 more than that of Q, what is R’s share in it
Sol: Let P = 2x , Q = 3x and R=5x. Now P+R-Q = 400 2x+5x-3x = 400 hence x =1OO R = 5x = 500.
27. Pencils, Pens and Exercise books in a shop are in the ratio of 10: 2 : 3. If there are 120 pencils, the number of exercise books in the shop is:.
Sol: Let Pencils = 10x, Pens = 2x & Exercise books = 3x. Now, 10x = 120 hence x = 12.
Number of exercise books = 3x = 36.
28. If p : q = 3 : 4 and q : r= 8 : 9, then p : r is
Sol: p/r = (p/q) * (q/r) = (3/4) * (8/9) = 2/3 so p : q = 2:3
29. Rs. 120 are divided among A, B, C such that A’s share is Rs. 20 more than B’s and Rs. 20 less than C’s. What is B’s share..
Sol: Let C = x. Then A = (x—20) and B = (x—40).
x + x - 20 + x - 40 = 120 Or x=60.
A:B:C = 40:20:60 = 2:1 :3.
Bs share = Rs. 120*(1/6) = Rs. 20.
30. If three numbers in the ratio 3 : 2: 5 be such that the sum of their squares is 1862, the middle number will be:
Sol: Let the numbers be 3x, 2x and 5x. Then,
9x + 4x + 25x =1862 ⇒ 38x = 1862 ⇒ x = 49 ⇒ x = 7.
middle number = 2x = 14.
31. In a college, the ratio of the number of boys to girls is 8 : 5. If there are 160 girls, the total number of students in the college is:
Sol: Let the number of boys and girls be 8x and 5x.
Total number of students = 13x = 13 x 32 = 416.
32. X, Y and Z share a sum of money in the ratio 7 : 8 : 16. If Z receives Rs. 27 more than X, then the total money shared was:
Sol: Let X = 7x, Y = 8x & Z = 16x. Then, total money = 31x.
Now, Z - X = 27 so 16x—7x = 27 that is why x = 3.
Total money 31*x = Rs.93.
33. An amount of money is to be distributed among F, Q and R in the ratio 3 : 5 : 7. If Qs share is Rs. 1500, what is the difference between Ps and Rs shares?.
Sol: Let P = 3x, Q = 5x and R = 7x.
Then, 5x = 1500 ⇒ x = 300. P=900,Q=1500 and R = 21OO.
Hence, (R - p) = (2100 - 900) = 1200
34. A profit of Rs. 30000 is to be distributed among A, B, C in the proportion 3 : 5 : 7. What will be the difference between B’s and C’s shares?
Sol: Bs share = Rs. 30000 *(5/15) = Rs.10000.
C’s share = Rs. 30000 * (7/15) = Rs.14000,
Difference in Bs and Cs shares = Rs.4000.
35. The compounded ratio of (2 : 3), (6: 11) and (11 :2) is
Sol; Required ratio = (2/3) * () * (6/11) * (11/2) = 2/1
36. The ratio of the number of boys and girls in a college is 7: 8. If the percentage increase in the number of boys and girls be 20% and 10% respectively, what will be the new ratio?
Sol: Originally, let the number of boys and girls in the college be 7x and 8x respectively.
Their increased number is (120% of 7x) and (110% of 8x).
120
x 7x
and
110
x 8x
100
100
42x
and
44x
5
5
The required ratio =
42x
:
44x
= 21: 22.
5
5
37. Salaries of Ravi and Sumit are in the ratio 2 : 3. If the salary of each is increased by Rs. 4000, the new ratio becomes 40 : 57. What is Sumit's salary?
Sol: Let the original salaries of Ravi and Sumit be Rs. 2x and Rs. 3x respectively.
Then,
2x + 4000
=
40
3x + 4000
57
57(2x + 4000) = 40(3x + 4000)
6x = 68,000
3x = 34,000
Sumit's present salary = (3x + 4000) = Rs.(34000 + 4000) = Rs. 38,000
38. The sum of three numbers is 98. If the ratio of the first to second is 2 :3 and that of the second to the third is 5 : 8, then the second number is:
Sol: Let the three parts be A, B, C. Then,
A : B = 2 : 3 and B : C = 5 : 8 =
5 x
3
:
8 x
3
= 3 :
24
5
5
5
A : B : C = 2 : 3 :
24
= 10 : 15 : 24
5
B =
98 x
15
= 30.
49
39. If Rs. 782 be divided into three parts, proportional to : : , then the first part is
Sol: Given ratio = : : = 6 : 8 : 9.
1st part = Rs.
782 x
6
= Rs. 204
23
40. The salaries A, B, C are in the ratio 2 : 3 : 5. If the increments of 15%, 10% and 20% are allowed respectively in their salaries, then what will be new ratio of their salaries?
Sol: Let A = 2k, B = 3k and C = 5k.
A's new salary =
115
of 2k =
115
x 2k
=
23k
100
100
10
B's new salary =
110
of 3k =
110
x 3k
=
33k
100
100
10
C's new salary =
120
of 5k =
120
x 5k
= 6k
100
100
New ratio
23k
:
33k
: 6k
= 23 : 33 : 60
10
10
41.If 40% of a number is equal to two-third of another number, what is the ratio of first number to the second number?
Sol: Let 40% of A =
2
B
3
Then,
40A
=
2B
100
3
2A
=
2B
5
3
A
=
2
x
5
=
5
B
3
2
3
A : B = 5 : 3.
42. Two number are in the ratio 3 : 5. If 9 is subtracted from each, the new numbers are in the ratio 12 : 23. The smaller number is:
Sol: Let the numbers be 3x and 5x.
Then,
3x - 9
=
12
5x - 9
23
23(3x - 9) = 12(5x - 9)
9x = 99
x = 11.
The smaller number = (3 x 11) = 33.
43. In a bag, there are coins of 25 p, 10 p and 5 p in the ratio of 1 : 2 : 3. If there is Rs. 30 in all, how many 5 p coins are there?
Sol: Let the number of 25 p, 10 p and 5 p coins be x, 2x, 3x respectively.
Then, sum of their values = Rs.
25x
+
10 x 2x
+
5 x 3x
= Rs.
60x
100
100
100
100
60x
= 30
x =
30 x 100
= 50.
100
60
Hence, the number of 5 p coins = (3 x 50) = 150.
44.In a mixture 60 liters, the ratio of milk and water 2 : 1. If the this ratio is to be 1 : 2, then the quantity of water to be further added is:
Sol: Quantity of milk =
60 x
2
litres = 40 litres.
3
Quantity of water in it = (60- 40) litres = 20 litres.
New ratio = 1 : 2
Let quantity of water to be added further be x litres.
Then, milk : water =
40
.
20 + x
Now,
40
=
1
20 + x
2
20 + x = 80
x = 60.
Quantity of water to be added = 60 litres.
45.A sum of money is to be distributed among A, B, C, D in the proportion of 5 : 2 : 4 : 3. If C gets Rs. 1000 more than D, what is B's share?
Sol: Let the shares of A, B, C and D be Rs. 5x, Rs. 2x, Rs. 4x and Rs. 3x respectively.
Then, 4x - 3x = 1000
x = 1000.
B's share = Rs. 2x = Rs. (2 x 1000) = Rs. 2000.
46.Zinc and copper are melted together in the ratio 9 : 11. What is the weight of melted mixture, if 28.8 kg of zinc has been consumed in it?
Sol: For 9 kg zinc, mixture melted = (9 + 11) kg.
For 28.8 kg zinc, mixture melted =
20
9
x 28.8
kg = 64 kg.
47. Gold is 19 times as heavy as water and copper is 9 times as heavy as water. In what ratio should these be mixed to get an alloy 15 times as heavy as water?
Sol: G = 19W and C = 9W.
Let 1 gm of gold be mixed with x gm of copper to get (1 + x) gm of the alloy.
(1 gm gold) + (x gm copper) = (x + 1) gm of alloy
19W + 9Wx = (x + 1) x 15W
19 + 9x = 15(x + 1)
6x = 4
x =
2
3
Ratio of gold with copper = 1 :
2
3
= 3 : 2.
48 The prices of a scooter and a T.V. are in the ratio 7 : 5. If the scooter costs Rs. 8000 more than a T.V. set, then the price of a T.V. set is :
Sol: Let the prices of a scooter and a T.V. set be Rs. 7x and Rs. 5x respectively.
Then, 7x - 5x = 8000
2x = 8000
x = 4000
Price of a T.V. set = Rs. (7 x 4000) = Rs. 28000.
49.A fraction which bears the same ratio
1
27
that
3
11
does to
5
9
, is equal to:
Sol: Let x :
1
27
::
3
11
:
5
9
. Then, x x
5
9
=
1
27
x
3
11
x =
1
27
x
3
11
x
9
5
=
1
55
50.A and B are two alloys of gold and copper prepared by mixing metals in the ratio 7 : 2 and 7 : 11 respectively. If equal quantities of the alloys are melted to form a third alloy C, the ratio of gold and copper in C will be:
Sol: Gold in C =
7
9
+
7
18
units =
7
6
units. Copper in C =
2
9
+
11
18
units =
5
6
units.
Gold : Copper =
7
6
:
5
6
= 7 : 5.
51. The sides of a triangle are in the ratio
1
2
:
1
3
:
1
4
and its perimeter is 104 cm.
The length of the longest side is :
Ratio of sides =
1
2
:
1
3
:
1
4
= 6 : 4 : 3.
Largest side =
104 x
6
13
cm = 48 cm.
52.Two numbers are in the ratio 1 : 2, If 7 is added to both, their ratio changes to 3 : 5, The greatest number is:
Sol:
Let the numbers be x and 2x.
Then,
x + 7
2x + 7
=
3
5
5(x + 7) = 3(2x + 7)
x = 14.
Greatest number = 28.
53. If 10% of x = 20% of y, then x : y is equal to:
Sol:
10% of x = 20% of y
10x
100
=
20y
100
x
10
=
y
5
x
y
=
10
5
=
2
1
x : y = 2 : 1.
54. If 0.75 : x :: 5 : 8, then x is equal to:
Sol:
(x x 5) = (0.75 x 8)
x =
6
5
= 1.20.
55. The ages of A and B are in the ratio of 3 : 1. Fifteen years hence, the ratio will be 2: 1. Their present ages are :
Sol;
Let the ages of A and B be 3x years and x years respectively.
Then,
3x + 15
x + 15
=
2
1
2x + 30 = 3x + 15
x = 15
So, A's age = (3 x 15) years = 45 years and B's age = 15 years.
56. If A : B : C = 2 : 3 : 4, then
A
B
:
B
C
:
C
A
is equal to:
Sol:
Let A = 2x, B = 3x and C = 4x. Then,
A
B
=
2x
3x
=
2
3
,
B
C
=
3x
4x
=
3
4
and
C
A
=
4x
2x
=
2
1
A
B
:
B
C
:
C
A
=
2
3
:
3
4
:
2
1
= 8 : 9 : 24.
57. If a carton containing a dozen mirrors is dropped, which of the following cannot be the ratio of broken mirrors to unbroken mirrors?
Sol:
For dividing 12 into two whole numbers, the sum of the tatio terms must be a factor of 12.
So, they cannot be in the ratio 3 : 2.
58.The ratio of the incomes of A and B is 5 : 4 and the ratio of their expenditures is 3 : 2. If at the end of the year, each saves Rs. 1600, then the income of A is :
SoL;
Let the incomes of A and B be Rs. 5x and Rs. 4x respectively
and let their expenditures be Rs. 3y and Rs. 2y respectively.
Then, 5x - 3y = 1600 ....(i) and 4x - 2y = 1600 ....(ii)
On multiplying (i) by 2, (ii) by 3 and subtracting, we get : 2x = 1600
x = 800.
A's income = Rs. 5x = Rs. (5 x 800) = Rs. 4000.
59. If A : B = 2 : 3, B : C = 4 : 5 and C : D = 6 : 7, then A : B : C : D is:
Sol: A : B = 2 : 3, B : C = 4 : 5 =
4 x
3
4
:
5 x
3
4
= 3 :
15
4
and C : D = 6 : 7 =
6 x
15
24
:
7 x
15
24
=
15
4
:
25
8
A : B : C : D = 2 : 3 :
15
4
:
35
8
= 16 : 24 : 30 : 35.
60. A sum of Rs. 1300 is divided amongst P, Q, R and S such that
P's share
Q's share
=
Q's share
R's share
=
R's share
S's share
=
2
3
. Then, P's share is :
Sol:
Let P = 2x and Q = 3x. Then,
Q
R
=
2
3
R =
3
2
Q =
3
2
x 3x
=
9x
2
Also,
R
S
=
2
3
S =
3
2
R =
3
2
x
9x
2
=
27x
4
Thus, P = 2x, Q = 3x, R =
9x
2
and S =
27x
4
Now, P + Q + R + S = 1300
2x + 3x +
9x
2
+
27x
4
= 1300
(8x + 12x + 18x + 27x) = 5200
65x = 5200
x =
5200
65
= 80.
P's share = Rs. (2 x 80) = Rs. 160.
61. A certain amount was divided between A and B in the ratio 4 : 3. If B’s share was Rs. 4800, the total amount was:
Sol: If B’s share is Rs. 3, total amount = Rs. 7.
If B’s share is Rs. 4800. total amount = Rs.
7
3
x 4800
= Rs. 11200.
62. A sum of Rs. 53 is divided among A, B, C in such a way that A gets Rs. 7 more than what B gets and B gets Rs. 8 more than what C gets. The ratio of their shares is:
Sol: Suppose C gets Rs. x.Then, B gets Rs. (x + 8) and A gets Rs. (x + 15).
Then, x + (x + 8) + (x + 15) = 53
x = 10.
A : B : C = (10 + 15) : (10 + 8) : 10 = 25 : 18 : 10.
63. The ratio of three numbers is 3: 4: 7 and their product is 18144. The numbers are:
Sol: Let the numbers be 3x, 4x and 7x. Then,
3x x 4x x 7x = 18144
x3 = 216,
x3 = 63
x = 6.
The numbers are 18, 24 and 42.
64. what least number must be subtracted from each of the numbers 14, 17, 34 and 42 so that the remainders may be proportional?
Sol: Let the required number be x.
Then, (14 – x) : (17 – x) : : (34 – x) : (42 – x).
14 - x
17 - x
=
34 - x
42 - x
(14 - x) (42 - x) = (17 - x) (34 - x)
=
x2 - 56x + 588
=
x2 - 51x + 578
5x = 10
x = 2
Required number = 2.
65. If 76 is divided into four parts proportional to 7, 5, 3, 4. then the smallest part is
Sol: Given ratio = 7 : 5 : 3 : 4, Sum of ratio terms = 19.
Smallest part =
76 x
3
19
= 12.
66.The ratio of the number of boys and girls in a school is 3 : 2. If 20% of the boys and 25% of the girls are scholarship holders, what percentage of the students does not get the scholarship?
Sol: Let boys = 3x and girls = 2x
Number of those who do not get scholarship = (80% of 3x) + (75% of 2x)
=
80
100
x 3x
+
75
100
x 2x
=
39x
10
Required percentage
=
39x
10
x
1
5x
x 100
% = 78%.
67.Two numbers are respectively 20% and 50% more than a third number. The ratio of the two numbers is:
Sol: Let the third number be x.
Then, first Number = 120% of x =
120x
100
=
6x
5
;
second number = 150% of x =
150x
100
=
3x
5
;
Ratio of first two numbers =
6x
5
:
3x
2
= 12x : 15x = 4 : 5.
68. If
A
3
=
B
4
=
C
5
, then A : B : C is :
Sol:
Let
A
3
=
B
4
=
C
5
= k. Then, A = 3k, B = 4k and C = 5k
A : B : C = 3k : 4k : 5k = 3 : 4 : 5.
69. An amount of Rs. 2430 is divided among A, B and C such that if their shares be reduced by Rs. 5, Rs.10 and Rs. 15 respectively, the remainders shall be in the ratio of 3 : 4 : 5. Then, B’s share was
Sol:
Remainder = Rs. [2430 – (5 + 10 + 15)] = Rs. 2400.
B's share = Rs.
2400 x
4
12
+ 10
= Rs. 810.
70.If
a
3
=
b
4
=
c
7
, then
a + b + c
c
is equal to :
Sol:
Let
a
3
=
b
4
=
c
7
= k, Then, a = 3k, b = 4k, c = 7k.
a + b + c
c
=
3k + 4k + 7k
7k
=
14k
7k
= 2.
71.The lease whole number which when subtracted from both the terms of the ratio 6 : 7 gives a ratio less than 16 : 21 is:
Sol:
Let x be subtracted. Then,
6 - x
7 - x
<
16
21
21 (6 - x) < 16 (7 - x)
5x > 14
x > 2.8
Least such whole number is 3.
72. The speeds of three cars are in the ratio 5 : 4 : 6. The ratio between the time taken by them to travel the same distance is :
Sol:Ratio of time taken =
1
5
:
1
4
:
1
6
= 12 : 15 : 10
73.In a school, 10% of the boys are same in number as
1
4
th of the girls.
.What is the ratio of boys to girls in that school?
Sol:
10% of B =
1
4
G
10B
100
=
1
4
G
B =
5
2
G
B
G
=
5
2
B : G = 5 : 2.
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