a) 60 paise & &1.35. 4:9
b) 4 months & 1 year 6 months. 2:9
c) 60cm & 2.4m. 1:4
d) 1.5kg & 600gms. 5:2
) Which ratio is greater - 17:21 or 23:28? 23:28> 17:21
) Arrange the following ratios in the ascending order:
A) 19:21, 13:15, 11:24. 11:24, 13:15 , 19:21
) Arrange the following ratios in the ascending order:
7:12,11:16,13:18. 13:18, 11:16, 7:12
) Which is Greater ratio (x + 4y)/(x + 5y) or (x + 5y)/(x +6y) ? (x + 5y)/(x +6y)
) Find the following:
a) Duplicate ratio of 2√a : 3√b. 4a:9b
b) Triplicate ratio of 2x: 3y. 8x³:27y³
c) sub duplicate ratio of 16x²: 25y². 4x: 5y
d) sub-triplicate ratio of 27a³: 125b³. 3a: 5b
e) The reciprocal ratio of 9:11. 11:9
*** Compound ratio of
a) duplicate ratio of 5:6, the reciprocal ratio of 25:32 and sub-duplicate ratio of 81:64. 1:1
b) x: 3y and 2a : b/3. 2ax: yb
c) x²/2 : y²/3 and the duplicate ratio of 1/x : 1/y. 3:2
d) 3a: 4b and the sub-duplicate ratio of 4/9a² : 1/16b². 2:1
e) 2y²: 3x² and the triplicate ratio of x : y. 2x: 3y
f) duplicate ratio of 5:6, reciprocal ratio of 25:42 and the sub-triplicate ratio of 216:343. 1:1
g) If x: y be the sub-duplicate ratio of (x - a): (y - a), prove that a = xy/(x +y).
) If (3x -9):(5x+4) is the triplicate ratio of 3:4, find the value of x. 12
) If A: B= 4:9 and B: C= 6:5, find A: C. 8: 15
) If A: B= 3:4 and B: C= 6:5, find A: B:C. 9: 12:10
) If A: B= 2:3 and B: C= 4:7, find A: B:C. 8: 12:21
) If A: B= 1/4:1/5 and B: C= 1/7:1/6, find A: B:C. 15: 12:14
) 3A =4 B = 6C, find A: B : C. 4:3:2
) If (2x +5y): (5x -7y)= 5:3, Find x: y. 50:19
) If (4x +3y): (6x +5y)= 11:17, Find x: y. 2:1
) If a: b= 2:3 find (5a+ 8b): (6a - 7b). 49:9
) if (6x² - 3y²):(x² + y²) is equal to 6: 25 find the value of x:y . 3:4
) What number must be added to each term of the ratio 3:5 to make it 11: 12? 19
) What same number should be added to both terms of 7:12 to get the ratio 2:3? 3
)
) The ages of A and B are in the ratio 7:8. 6 years ago, their ages were in the ratio 5:6. Find their present ages. 21,24yrs
) A ratio is equal to 3:4. If its consequent is 144, what is its antecedent? 108
) Two numbers are in the ratio 5: 7. If 8 is subtracted from each, the ratio becomes 3:5. Find the numbers. 20,28
) What should be subtracted from each term of the ratio 4:9 so that it becomes 1:3. 3/2
) Two numbers in the ratio 3:5. If 8 is added to each number, the ratio becomes 2:3. Find the numbers. 24,40
) Find the number which bears the same ratio to 7/33 that 8/21 does to 4/9. 2/11
) Work done by (x- 3) men in (2x + 1) days and the work done by (2x + 1) men in (x + 4) days are in the ratio 3:10. Find the value of x. 6
) the cost of production of radio set is ₹900, which is divided between material, labour and overheads in the ratio 3:4:2. Calculate the cost of material used in the radio set. 300
)
) A sum of money is divided in the ratio of 3:5. If the larger part is ₹3125, find the smaller part. 1875
) Divide ₹2600 among A, B and C in the ratio 1/2: 1/3: 1/4. 1200,800,600
) when the fair of a certain journey by an airline was increased in the ratio 5:7, the cost of a ticket for the journey becomes ₹1421. Find the increase in the fare. 406
)
)
)
) Sum of the two numbers is 38 and their difference is 2. What is the ratio between the numbers? 10:9
) the ratio of the males to females in a committee of 48 members is 31. How many more ladies b added to the committee so that the ratio of the males to females may be 9:5 ? 8
) a bav contains ₹142 in the form of Rs 1, 50 paise and 20 paise coins in the ratio of 3:5:8. Find the number of coins of each type. 60,100,160
) A, B and C play cricket. The rund scored by A and B respectively are in the ratio 3:2. B's runs to C's runs are also in the ratio 3:2, and together they score 342 runs. How many runs did each score ? 162, 108, 72
) Ages of two persons A and B are in the ratio 4:3. Five years hence the ratio of their ages will change to 9:7. Find their present ages. 40, 30yrs
) In a mixture of 45 litres, the ratio milk to water is 13:2. How much water must be added to this mixture to make the ratio of milk to water as 3:1? 7 litres
) In an examination, the ratio passes to failure was 4 :1. Had 30 less appeared and 20 less passed, the ratio of passes to failure would have been 5:1. How many students appeared for the examination ? 150
PROPORTION
1) *** Find x, when
a) If x², 4 and 9 are in continued proportion. 4/3
b) 1:3:: x:7. 7/3
c) x: 1.5 ::3:5 . 0.9
d) 2.5 :1.5:: x :3. 5
e) x: 50:: 3:2 . 75
) Find the third proportional to:
a) 9, 6. 4
b) 8/3, 4. 6
c) 1.6, 2.4. 3.6
d) 0.24, 0.6. n. ....1.5
e) (2 + √3),(5 + 4√3). (26+7√3)
f) [a/b + b/a] and √(a²+b²). ab
3) Find the fourth proportional to:
a) 3, 12 and 15. 60
b) 1.4, 3.2 and 7. 16
c) 1.5, 4.5 and 3.6. 10.8
d) (a²- ab +b²),(a³+ b³) and (a - b). a²- b²
4) Find the mean proportion of :
a) 5 and 80. 20
b) 2.5 and 0.9. 1.5
c) 1 / 12 and 1 / 75. 1/30
d) (8 - 4√3) and (6 + 3√3). 2√3
) What number should be subtracted from each of the following numbers 23, 30, 57 and 78 so that the reminders are in proportion ? 6
)
) what least number must be adde to each of the number 16, 7, 79 and 43, so that the resulting numbers are in proportion ? 5
) Find two numbers whose mean proportional is 16 and the third proportion is 128. 8,32
) What number should be added to each of the number 12, 22, 42 and 72 so that the resulting number maybe proportion. 3
) if (a²+ c²)(b²+ d²(= (ab + cd)² prove that a, b, c, d are in proportion.
)
) If (x - 2), (x + 2), (2x + 1) and (2x + 19) are in proportion, find the value of x. 4
) if (x + 5) is the geometric mean between (x+ 2) and (x + 9), find the value of x. 7
) If b is the mean proportional between a and c, prove:
a) abc(a+b+c)³= (ab+ bc+ c a)³.
) If x/a = y/b = z/c Prove:
a) [(a²x²+ b² y²+ c² z²)/(a³x + by³+ c³z)]³⁾² = √(xyz/abc).
b)
) If a/b = c/d = e/f, prove that:
a) (a²+d² + f²)/(a²+ c² + e²) = (ab + cd + ef)².
b) (a³+c³)²/(b³+ d³)² = e⁶/f⁶.
c) (pa³+qc³+ re³)/(pb³+ qd³+ rf³) = ace/bdf
e) a²/b² + c²/d² + e²/f² = ac/bd + ce/df + ae/bf.
) a, b, c, d are in continued proportion, prove that :
a) (a-b)³/(b-c)³ = a/d
b) √(ab) + √(bc) - √(cd) = √{(a+ b- c)(b + c - d)}.
c) (a+ b)/(c+ d) =√(2a²+7b²)/√(2c²+ 7d²)
d) (ma²+nc²)/(mb²+ nd²) = √(a⁴+b⁴)/√(b⁴+ d⁴)
e) (a² + ab+b²)/(a²- ab +b²) =(c²+cd+ d²)/(c²- cd + d²)
f) (a+c)³/(b+ d)³ = a(a - c)²/b(b - d)².
) If ax =by = cz, Prove x²/yz + y²/zx + z²/xy = bc/a² + ca/b² + ab/c².
EXERCISE
*** Prove the following as a :b: : c: d or a/b = c/d
) (4a+5b)(4c-5d) = (4a-5b)(4c+5d)
) (pa+qb):(pc+qd) :: (pa-qb):(pc- qd)
) (5a+6b)/(5c+6d)=(5a-6b)/(5c-6d).
) (3a+4b)/(3c+4d)=(3a-4b)/(3c-4d).
) (a+3b+2c+6d)/(a-3b+2c-6d) = (a+3b-2c- 6d)/(a-3b-2c+6d)
) (a+b+c+d):(a+b-c-d) :: (a-b+c- d):(a-b-c+d).
) (2a+2b-3c-3d)/(2a-2b-3c+3d) = (a+b- 4c- 4d)/(a-b-4c+4d).
) If a: b:: c: d, Prove the following:
a) (2a+5b)/(2a-5b) = (2c+ 5d)/(2c-5d).
b) (5a+11b)/(5c+11d) = (5a-11b)(5c-11d)
c) (2a+3b)(2c-3d) = (2a-3b)(2c+3d)
) (3a - 5b)/(3c+5d)=(3c- 5d)/(3c+ 5d)
d) (la+mb):(lc+ md) :: (la-mb): (lc- md).
) If x= 2ab/(a + b) , Prove that (x+a)/(x -a) + (x+ b)/(x-b) = 2
) If x= 6ab/(a + b) , Prove that (x+3a)/(x -3a) + (x+ 3b)/(x- 3b) = 2
) If (x³+3x)/(3x²+1) = 341/91, Prove that x= 11
** Find x from the following:
a) {√(2- x) + √(2+x)}/{√(2- x) - √(2+x)}= 3. -6/5
b) {√(1+ x) + √(1-x)}/{√(1+ x) - √(1- x)}= a/b. 2ab/(a²+ b²)
c) {√(a+ x) + √(a-x)}/{√(a+ x) - √(a- x)}= c/d. 2acd/(c²+ d²)
) {√(x+4) + √(x -10)}/{√(x +4) - √(x -10)}= 5/2. 263/20
d) {(a- x)/(a+x)}³ =16{(a+ x)/(a-x)}. -a/3, -3a
e) (1+ x+ x²)/(1-x+ x²)=62(1+x)/63(1-x). 1/5
f) {√(x +5) + √(x -16)}/{√(x+5) - √(x -16)}= 7/3. 20
Prove
a) If {√(2a +1) + √(2a -1)}/{√(2a+1) - √(2a -1)}= x. Then x²- 4ax +1= 0
b) {√(b +3a) + √(b -3a)}/{√(b+3a) - √(b -3a)}= x, then 3ax²- 2bx + 3a= 0.
c) {3√(m +1) + ³√(m -1)}/{³√(m+1) - ³√(m -1)}= x then x³- 3x²m+ 3x - m= 0.
d) {√(a +3b) + √(a -3b)}/{√(a+3b) - √(a -3b)}=x, then 3by²- 2ax + 3b = 0
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