Exercise -1
A) find the third proportion to
1) 6, 30. 150
2) 8, 12. 18
3) a/b + b/a and √(a²+b²). ab
4) 5+ 2√3 and 37+20√3 305+ 17√3
B) Find the fourth proportional to:
1) a+1, a+2, a²+3a+2. (a+2)²
2) x²-4x+3, x²+x-2, x²-9. x²+5x+6
3) p²-pq+q², p³+q³, p-q. p²-q²
4) 8, 13, 16. 26
C) Find the mean proportional between:
1) 6; 54 18
2) (a+b)(a-b)³; (a+b)³(a-b). (a²-b²)²
3) (x-y); (x³- x²y). x² - xy
4) √27- 3√2 and√27+3√2. 3
D) If a: b:: b: c, prove a:c= a²:b²
E) If y is the mean proportional between x and z, prove that xy+ yz is the mean proportional between x²+y² and y²+z².
F) Find x, if
1) x:3= 2:1. 6
2) 7:x= x: 343. 49
3) 3:18= x:36. 6
4) 7:35= 6:x. 30
G) Find two numbers numbers such that the mean proportional between them is 14 and the third proportional to them is 112. 7
H) What must be added to each of the four numbers numbers 10, 18, 22, 38 so that they become in proportion. 2
I) What must be subtracted from each of the numbers 21, 38, 55, 106 so that they become in proportion. 4
J) Find two numbers such that the mean proportion between them is 24 and the third proportional to them is 192. 12, 48
K) Find the number which must be added to each of the number 15, 17, 34 and 38 so that they may become in proportion. 4
L) If three quantities are in continued proportion, prove that the first is to the third the third is the duplicate ratio of the first to the to the second.
M) if a≠ b and a:b is the duplicate ratio of a+c and b+c, prove that c is the mean proportion between a and b.
N) What must be added to the number 6, 10, 14 and 22 so that they become proportional. 2
O) Find the two numbers such that their mean proportional is 24 and the third proportional is 1536. 6,96
P) If x and y are unequal and x:y is the duplicate ratio of x+y and y+z, prove that z is mean proportional between x and y
Q) If q is the mean proportional between p and r, prove that p² - q²+ r² = q⁴(1/p² - 1/q² + 1/r²)
R) If b is the mean proportional between a and c show that abc(a+b+c)³ = (ab +bc+ ca)³
P2)
A) If a, b, c, are in continued Proportion, Prove that
1) a:d:: pa³+qb³+ rc³: pb³+ qc³+rd³
2) a-b: a+b:: a-d: a+2b+2c+d
3) (b-c)²+ (c-d)²+(d-b)²= (a-d)²
4) a+b: c+d:: √(a²+b²):√(c²+d²)
5) a:d:: (pa³+qb³+rc³):(pb³+qc³+rd³)
6) (a-b):(a+b)::(a-d):(a+2b+2c+d)
7) (b-c)²+(c-a)²+(d-b)²=(a-d)²
8) √{(a+b+c)(b+c+d)}= √(ab) +√(bc) + √(cd)
9) √[{(a+b+c)(b+c+d)}/{√(ab)+√(bc) + √(cd)}] =1
10) (a²+b²+c²)(b²+c²+d²)= (ab+bc+cd)²
11) a/d = (a-b)³/(b-c)³
12) √{(a+b+c)(b+c+d)}= √(ab)+ √(bc) + √(cd)
13) a³+c³+e³: b³+d³+f³:: ace: bsf
14) 4(a+b)(c+d)= bd[(a+b)/b + (c+d)/d]²
15) (ab+cd)²= (a²+c²)(b²+d²)
16) a²+b²: a² - b²:: a+c : a- c
17) (a+b+c)(a-b-c)= a²+b²+c²
18) (a+b+c)²/(a²+b²+c²)= (a+b+c)/(a-b+c)
19) (a+b):(b+c):: a²(b-c): b²(a-b)
20) (a+b+c)²:(a²+b²+c²):: (a+b+c)(a-b+c)
B) If x/a= y/b = z/c, show that:
1) {(a²x²+ b²y²+ c²z²)/(a³x+b³y+c³z)}³⁾²= √(xyz/abc)
2) (x²+y²+z²)/(a²+b²+c²) = {(px+qy+rz)/(pa+ab+rc)}²
3) x³/a³ - y³/b³+z³/c³= xyz/abc
4) (ax-by){(a+b)(x-y)} + (by- cz)/ {(b+c)(y-z)} + (cz-ax)/{(c-a)(z-x)} = 3
C)
If a/b = c/d = e/f, Prove that
1) (ab+cd+ef)²=(a²+c²+e²)(b²+d²+ f²)
2) (a+3c-5e)/(b+3d-5f) is...
3) ³√{(a³-2c³+3e³)/(b³-2d³+3f³)}is.
4) √{(a²+c²+e²)/(b²+a²+f²)
5) (b²+d²+f²)(a²+c²+e²)= (ab+cd+ef)²
6) (a+c+e)/(b+d+f) is...
7) (a³+c³+e³)/(b³+d³+f³)=ace/bsf
8) {(a²b²+c²d²+e²f²)/(ab³+cd³+ef³)}³⁾²=√(ace/bsf)
Solve:
1) (1-px)/(1+px). √{(1-qx)/(1+qx)}= 1. 0, ±1/p √(2p-q)/q
2) {a+√(a²-2ax)}/{a- √(a²- 2ax)}= b. x= 2ab/(b+1)²
1) If x/(b+c-a) = y/(c+a-b) = z/(a+b-c), show that (b--c)x + (c-a)y +(a-b)z =0
Exercise -2
1) Find the third proportional to 15, 20.
2) Divide ₹1320 among 7 men, 11 women & 5 boys so that each women may have 3 times as much as a boy, and a man as much as a woman and a boy together. Find how much each person receives.
3) How many one rupees coins, fifty-paise coins & twenty-five paise coins of which the numbers are proportional to 5/1, 3 and 4 are together worth & 210 ?
4) In 40 litres mixture of milk and water in 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 ?
5) Side of a hexagon becomes 3 times. Find ratio of areas of new and old hexagons.
6) find the number which when added to the terms of the ratio 11: 23 makes it equal to the ratio 4:7.
7) the sum of two numbers is 84. if the two numbers are in the ratio 4:3, find the two numbers. 48, 36
8) if 4a = 3b, find (7a+9b):(4a+5b). 57: 32
9) the number of red balls and green balls in a bag are in the ratio 16:7. if there are 45 more red balls then green balls. find the number of green balls in the bag. 45
10) what least number must be added to each of a pair of numbers which are in the ratio 7:16 so that the ratio between the terms become 13:22 ? 6
11) A number is divided into four parts such that 4 times the first part, 3 times the second part, 6 times the third part and 8 times the fourth part are equal. In what ratio is the number divided? 6:8:4:3
12) Divide 3150 into four parts such that half of the first part, a third of the second part, a fourth of the third part is equal to one- twelfth of the fourth part. 300, 450, 600,1800.
13) if x: y= 4:3, y: z= 2:3, find x:y:z. 8:6:9
14) If a/b = 4/5, then find (2a²-3b)/(7a+6b²). Can't determined
15) Two numbers are in the ratio 4:5, if 7 is added to each, the ratio between the numbers becomes 5: 6. find the numbers. 28, 35
16) Find x: if x+2 : 4x+1 :: 5x+2 : 13x+1. 0 or 2.
17) The total monthly sales of two companies A and B are in the ratio 2:3 and their total monthly expenditures are in the ratio 3:4. Find the ratio of the profits of the two companies given that company A's profit is equal to a fifth of its sales. 6:13
18) If a: b= 3:7, what is the value of (4a+5b)/(2a+2b).
A) 47/20 B) 36/24 C) 56/32 D)10/4 E) none.
19) If a: b= 2:3, b: c= 4:3 , and If c: d= 2:3, find a: b: c: d.
A) 8:12:9:27 B) 16:24:18:27
C) 18:27:36:8 D) none
20) Vipin's present is twice what the age of Kishor was one year ago. What is the sum of their present ages (in years), if the ratio of the sum of their present ages to the difference of their present ages is 19:5?
A) 21 B) 19 C) 24 D) 34
21) The weight of the Bimal and Basu are in the ratio 2:3 and the weights of Basu and Bali are in the ratio 4:3. what is Basu's weight (in kg) if the sum of the weights of Bimal, Basu and Bali is 203 kgs ?
A) 84 B) 76 C) 49 D) 65 E) none
22) the ratio of the number of boys to the number of girls in school is 7:3. If an additional 15 girls were to join the class, the ratio of the number of boys to the number of girls would become 2:3. what is the initial number of the girls in the class ?
A) 4 B) 6 C) 12 D) 5 E) none
23) if k= (a+c)/(b+d) = (c+r)/(f+d) = (a+e)/(b+f) when all quantities are positive, then which of the following must be true ?
A) k= e/f B) k= a/b C) k= c/d D) All of the above E) none
24) if 3 is subtracted from the numerator and 5 is added to the denominator of a fraction, the new fraction formed is 1/2. if 2 is added to the numerator of the initial fraction, the ratio of the new numerator to the denominator becomes 1:1. find the original fraction.
A) 11/13 B) 18/23 C) 13/15 D) 5/14 E) none
25) The ratio of the Ahmed's age to Mohammad's age is the same as the ratio of the ages of their respective elder brothers. The ratio of the difference of the ages of the ages of Ahmed and Mohammad to that the difference of the ages of their respective brothers is 1:2. what is the ratio of the sum of the ages of their respective brothers to the sum of the ages of the Ahmed and Mohammad?
A) 2:1 B) 4:1 C) 3:1 D) √2:1
26) Three different types of balls priced at ₹5, ₹8 and ₹13 per piece are displayed in three different boxes by a trader. Mr Paul bought from this shop all three types of balls spending a total sum of ₹768. The numbers of the balls ihe bought, taken in the order in which the prices are mentioned above, are in the ratio 5:4:3. How many balls of the costliest veriety did he buy ?
A) 104 B) 64 C) 24 D) 72
27) the mean proportional between two numbers is 9 and third proportion of the two numbers is 243. find the larger of the two numbers.
A) 27 B) 81 C) 9 D) none
28) If 3x - 4y +2z= 0 and 4x - 2y-z= 0, find x:z: y.
A) 8:10:11 B) 8:11:40
C) 11:40:8 D) 8:40:11
29) A person with him a certain number of weighing stones of 100gms, 500 gms and 1kg in the ratio 3:5:1. If a maximum of 5 kg can be measured using weighing stones of 500 gms alone, then what is the number of 100 gm stones he has ?
A) 6 B) 3. C) 9 D) 5
30) What must be subtracted from p and added to q so that the ratio of the resultant becomes 1:3.
A)(p+q)/3 B) (3p-q)/4 C) (p-q)/(p+q) D) (q-3p)/4
31) p,q,r,s, n and m are positive integers such that 4p= 5q= 6r= 12s= 8n = 9m. Which of the following pairs contains a number, which is not integer?
A) {pq/54, (m+n)/17}
B) {q/9, (rs/p}
C) {(r-s)(n-m),(q+s)/17}
D) {p/m, m/q}
** the amount used to purchase one litre of petrol can be used to purchase 3 litres of diesel of 5 litres of kerosene. Out of certain amount, ₹510 is spent on diesel.
32) how much is spent on kerosene if equal volumes of the 3 liquids are purchased with the total amount ?
A) ₹300 B) ₹306 C) ₹382 D) ₹354
33) what will be the amount spent on petrol if the total amount referred in Q. 56 is spent to purchase equal volumes of petrol and kerosene only ?
A) 1250 B)1275 C)1955 D) 1360
34) some apples are divided among 4 people Karan, Kiran, Kumar and Khanna. the ratio of the number of apples given to Kiran to the total number of apples given to Karan and Khanna is 1:2. the ratio of the number of apples given to Kumar to that of the remaining apples is 2:5. Khanna gets 2 apples more than Kiran, Karan gets half the number of apples that Kumar get. What is the total number of apples distributed ?
A) 21 B) 18 C) 19 D) 24
35) If (2x²-4x+3)/(4x-3) =(2x²-3x+5)/(3x-5) Find the value/s of x.
A) 0 B) 2 C) -2 D) both A and C
36) A certain number is added to each of a pair of a numbers which are in the ratio 4:7. The sum of the resulting numbers is 75 and their ratio (taken in the same order as mentioned above) is 8:17. What is the number added ?
A) -12 B) 9 C) -13 D) 8
37) A garrison of 900 soldiers had food-stock sufficient for 30 days when the rate of consumption is 2.5 kg/day/soldier. After some days of consumption at the rate, 300 soldiers were transferred to another garrison and the balance food lasted for 25 days for the remaining soldiers. If the rate of consumption of the remaining soldiers was 3.0 kg/day/soldier, after how many days from the start, were the soldiers transferred?
A) 12 B) 10 C) 8 D) 15
38) Eight farmers take 4 hours to plough 12 acres of land. What is the area of land that will be ploughed, If 12 farmers work for 14 hours at the same efficiency as in the previous case ?
A) 52 acres B) 63 acres D) 58 acres D) 65 acres
39) Divide 66 into three parts such that the sum of the first two parts equals the third part and the second part is 3 less than twice the first part. What is the ratio of the parts are arranged in the ascending order ?
A) 14:17:19 B) 5:9:11 C) 11:7:4 D) 4:7:11
40) The speed of a locomotive without any wagons attached to it is 40 kmph. It diminishes by a quantity which is the proportional to the cube root of the number of wagons attached. if the speed of the locomotive is 34 kmph when 27 Wagon are attached, what is the maximum number of wagons that can be attached if the condition is that speed should not be fall below 30 kmph ?
A) 64 B) 125 C) 216 D) 343
41) A certain amount of money is divided among nine brothers. The second brother gets ₹2 more than twice the amount given to the first brother. the third brother gets ₹3 more than the thrice the amount given to the first brother, and so on till the ninth brother. If the ratio of the amount with the ninth brother to the amount with the first brother is 10:1, how much did the fifth brother get (in rupees)?
A) 100 B) 50 C) 90 D) 75
42) Manoj and Shiva, who are colleageues in an office, have their monthly savings in the ratio 2:3. Manoj spends two-thirds of his income every month. if the ratio of their monthly income is 3:4, what is the ratio of their expenditures?
A) 3:2 B) 4:5 C) 4:3 D) 5:3
** A test of 60 minutes contains questions on mathematics and English only. the time taken to solve a mathematics question is twice the time taken to answer an English question and the ratio of time taken to solve the mathematics questions to time taken to answer all English questions is 8/7.
43) what is the ratio of the number of English Questions to that of mathematics ?
A) 11/7 B) 7/4 C) 9/4 D) 7/5
44) if the total number of questions is 22. how many English questions can be answered in 18 minutes ?
A) 8 B) 10 C) 11 D) 9
