Applying componendo and dividendo: \[\begin{array}{*{35}{l}} 8a/18b\text{ }=\text{ }8c/18d \\ a/b\text{ }=\text{ }c/d \\ \end{array}\]
If x= (fig 1) ,find the value of:
fig 1: SOLUTION: \[\begin{array}{*{35}{l}} x\text{ }=\text{ }2ab/\text{ }\left( a\text{ }+\text{ }b \right) \\ x/a\text{ }=\text{ }2b/\left( a\text{ }+\text{ }b \right) \\ \end{array}\] Applying...
If x and y be unequal and x: y is the duplicate ratio of x + z and y + z, prove that z is mean proportional between x and y.
\[\begin{array}{*{35}{l}} x\left( {{y}^{2}}~+\text{ }{{z}^{2}}~+\text{ }2yz \right)\text{ }=\text{ }y\left( {{x}^{2}}~+\text{ }{{z}^{2}}~+\text{ }2xz \right) \\ x{{y}^{2}}~+\text{...
Find two numbers such that the mean proportional between them is 14 and third proportional to them is 112.
Let the required numbers be a and b. Given, 14 is the mean proportional between a and b. \[\begin{array}{*{35}{l}} a:\text{ }14\text{ }=\text{ }14:\text{ }b \\ ab\text{ }=\text{ }196 \\ a\text{...
Find the (iii) mean proportional to (x – y) and (x^3 – x^2y).
(iii) Let the mean proportional to \[\left( x\text{ }-\text{ }y \right)\text{ }and\text{ }\left( {{x}^{3}}~\text{ }-{{x}^{2}}y \right)\] be n. \[\left( x\text{ }-\text{ }y \right)\text{ }and\text{...
Find the: (i) fourth proportional to 2xy, x2 and y2. (ii) third proportional to a2 – b2 and a + b.
(i) Let the fourth proportional to 2xy, x2 and y2 be n. \[\begin{array}{*{35}{l}} 2xy:\text{ }{{x}^{2}}~=\text{ }{{y}^{2}}:\text{ }n \\ 2xy~\times ~n\text{ }=\text{ }{{x}^{2}}~\times ~{{y}^{2}} \\...
If 15(2×2 – y2) = 7xy, find x: y; if x and y both are positive.
15(2x2 – y2) = 7xy Let the substitution as \[\begin{array}{*{35}{l}} x/y\text{ }=\text{ }a \\ 2a\text{ }-\text{ }1/a\text{ }=\text{ }7/15 \\ \left( 2{{a}^{2}}~-\text{ }1 \right)/\text{ }a\text{...
A woman reduces her weight in the ratio 7: 5. What does her weight become if originally it was 84 kg?
Let the woman’s reduced weight as x. the original weight = 84 kg So, we have \[\begin{array}{*{35}{l}} 84:\text{ }x\text{ }=\text{ }7:\text{ }5 \\ 84/x\text{ }=\text{ }7/5 \\ 84\text{ * }5\text{...
What quantity must be added to each term of the ratio x: y so that it may become equal to c: d?
Let the required quantity which has to be added be p. \[\begin{array}{*{35}{l}} dx\text{ }+\text{ }pd\text{ }=\text{ }cy\text{ }+\text{ }cp \\ pd\text{ }-\text{ }cp\text{ }=\text{ }cy-\text{...
Find the value of x, if: (iii) (3x – 7): (4x + 3) is the sub-triplicate ratio of 8: 27.
(iii) (3x – 7): (4x + 3) is the sub-triplicate ratio of 8: 27 And the sub-triplicate ratio of \[\begin{array}{*{35}{l}} 8:\text{ }27\text{ }=\text{ }2:\text{ }3 \\ \left( 3x\text{ }-\text{ }7...
Find the value of x, if: (i) (2x + 3): (5x – 38) is the duplicate ratio of √5: √6. (ii) (2x + 1): (3x + 13) is the sub-duplicate ratio of 9: 25.
(i) (2x + 3): (5x – 38) is the duplicate ratio of √5: √6 And, the duplicate ratio of √5: √6 = 5: 6, \[\begin{array}{*{35}{l}} \left( 2x\text{ }+\text{ }3 \right)/\text{ }\left( 5x\text{ }-\text{ }38...
Find the : (v) reciprocal ratio of 3: 5 (vi) ratio compounded of the duplicate ratio of 5: 6, the reciprocal ratio of 25: 42 and the sub-duplicate ratio of 36: 49.
(v) Reciprocal ratio of \[3:\text{ }5\text{ }=\text{ }5:\text{ }3\] (vi) Duplicate ratio of \[5:\text{ }6\text{ }=\text{ }25:\text{ }36\] Reciprocal ratio of \[25:\text{ }42\text{ }=\text{...
Find the: (iii) sub-duplicate ratio of 9x^2a^4 : 25y^6b^2 (iv) sub-triplicate ratio of 216: 343
(iv) Sub-triplicate ratio of \[216:\text{ }343\text{ }=~{{\left( 216 \right)}^{1/3}}:\text{ }{{\left( 343 \right)}^{1/3}}~=\text{ }6:\text{ }7\] (v) Reciprocal ratio of \[3:\text{ }5\text{ }=\text{...
Find the: (i) duplicate ratio of 2√2: 3√5 (ii) triplicate ratio of 2a: 3b
(i) Duplicate ratio of \[2\surd 2:\text{ }3\surd 5\text{ }=\text{ }{{\left( 2\surd 2 \right)}^{2}}:\text{ }{{\left( 3\surd 5 \right)}^{2}}~=\text{ }8:\text{ }45\] (ii) Triplicate ratio of...
If (3x – 4y): (2x – 3y) = (5x – 6y): (4x – 5y), find x: y.
Since, \[\left( 3x\text{ }-\text{ }4y \right):\text{ }\left( 2x\text{ }-\text{ }3y \right)\text{ }=\text{ }\left( 5x\text{ }-\text{ }6y \right):\text{ }\left( 4x\text{ }-\text{ }5y \right)\] This...
If 5x + 6y: 8x + 5y = 8: 9, find x: y.
Given, On cross multiplying, we get \[\begin{array}{*{35}{l}} 45x\text{ }+\text{ }54y\text{ }=\text{ }64x\text{ }+\text{ }40y \\ 14y\text{ }=\text{ }19x \\ {} \\ x/y\text{ }=\text{ }14/19 \\...
If a: b = 3: 5, find: (10a + 3b): (5a + 2b)
\[\begin{array}{*{35}{l}} a/b\text{ }=\text{ }3/5 \\ \left( 10a\text{ }+\text{ }3b \right)/\text{ }\left( 5a\text{ }+\text{ }2b \right) \\ ~ \\ \end{array}\]
If (a + b + c + d) (a – b – c + d) = (a + b – c – d) (a – b + c – d), prove that a: b = c: d.
Rewriting the given, we have applying componendo and dividendo: Applying componendo and dividendo :
If a = 4√6/ (√2 + √3), find the value of:
Solution: Given, \[\begin{array}{*{35}{l}} a\text{ }=\text{ }4\surd 6/\text{ }\left( \surd 2\text{ }+\text{ }\surd 3 \right) \\ a/2\surd 2\text{ }=\text{ }2\surd 3/\text{ }\left( \surd 2\text{...
If x = 6ab/ (a + b), find the value of:
Solution: Given, \[\begin{array}{*{35}{l}} x\text{ }=\text{ }6ab/\text{ }\left( a\text{ }+\text{ }b \right) \\ \Rightarrow \text{ }x/3a\text{ }=\text{ }2b/\text{ }a\text{ }+\text{ }b \\...
If (7a + 8b) (7c – 8d) = (7a – 8b) (7c + 8d); Prove that a: b = c: d
If (fig 1) Then prove that x: y = u: v
SOLUTION: \[\begin{array}{*{35}{l}} 10x/\text{ }12y\text{ }=\text{ }10u/\text{ }12v \\ {} \\ x/y\text{ }=\text{ }u/v\text{ }\Rightarrow \text{ }x:\text{ }y\text{ }=\text{ }u:\text{ }v \\...
Given, a/b = c/d, prove that: (3a – 5b)/ (3a + 5b) = (3c – 5d)(3c + 5d)
If a : b = c : d, prove that: (6a + 7b) (3c – 4d) = (6c + 7d) (3a – 4b).
Since, a/b = c/d \[\left( 6a\text{ }+\text{ }7b \right)\left( 3c\text{ }-\text{ }4d \right)\text{ }=\text{ }\left( 3a\text{ }-\text{ }4b \right)\left( 6c\text{ }+\text{ }7d \right)\]
If a : b = c : d, prove that: (iii) xa + yb : xc + yd = b : d.
Since,a/b = c/d
If a : b = c : d, prove that: (i) 5a + 7b : 5a – 7b = 5c + 7d : 5c – 7d. (ii) (9a + 13b) (9c – 13d) = (9c + 13d) (9a – 13b).
(i) since, a/b = c/d (ii) since, a/b = c/d On cross-multiplication we have, \[\left( 9a\text{ }+\text{ }13b \right)\left( 9c\text{ }-\text{ }13d \right)\text{ }=\text{ }\left( 9c\text{ }+\text{ }13d...
What least number must be subtracted from each of the numbers 7, 17 and 47 so that the remainders are in continued proportion?
Let the number subtracted to be x. therefore, \[\begin{array}{*{35}{l}} \left( 7\text{ }-\text{ }x \right):\text{ }\left( 17\text{ }-\text{ }x \right)::\text{ }\left( 17\text{ }-\text{ }x...
If a/b = c/d, show that:
SOLUTION: Let a/b = c/d = k Therefore, a = bk and c = dk Taking L.H.S, Now, taking the R.H.S Thus, L.H.S = R.H.S
If a, b, c are in continued proportion and a(b – c) = 2b, prove that: a – c = 2(a + b)/ a
a, b, c are in continued proportion. So, \[\begin{array}{*{35}{l}} a/b\text{ }=\text{ }b/c \\ \Rightarrow \text{ }{{b}^{2}}~=\text{ }ac \\ \end{array}\] And, \[\begin{array}{*{35}{l}} a\left(...
If a, b, c are in continued proportion, show that
SOLUTION: a, b, c are in continued proportion. So, \[\begin{array}{*{35}{l}} a/b\text{ }=\text{ }b/c \\ \Rightarrow \text{ }{{b}^{2}}~=\text{ }ac \\ \end{array}\] \[\begin{array}{*{35}{l}} \left(...
What least number must be added to each of the numbers 6, 15, 20 and 43 to make them proportional?
Let the number added to be x. therefore, \[\begin{array}{*{35}{l}} \left( 6\text{ }+\text{ }x \right):\text{ }\left( 15\text{ }+\text{ }x \right)\text{ }::\text{ }\left( 20\text{ }+\text{ }x...
If x^2, 4 and 9 are in continued proportion, find x.
x2, 4 and 9 are in continued proportion So, we have \[\begin{array}{*{35}{l}} {{x}^{2}}/4\text{ }=\text{ }4/9 \\ {{x}^{2}}~=\text{ }16/9 \\ Thus,\text{ }x\text{ }=\text{ }4/3 \\...
If x + 5 is the mean proportional between x + 2 and x + 9; find the value of x. Solution:
x + 5 is the mean proportional between x + 2 and x + 9. So, (x + 2), (x + 5) and (x + 9) are in continued proportion. \[\begin{array}{*{35}{l}} \left( x\text{ }+\text{ }2 \right):\text{ }\left(...
Find the mean proportional between: (i) 6 + 3√3 and 8 – 4√3 (ii) a – b and a^3 – a^2b
(i) Let the mean proportional between \[6\text{ }+\text{ }3\surd 3~and\text{ }8\text{ }-\text{ }4\surd 3~\] be x. So, \[6\text{ }+\text{ }3\surd 3,\text{ }x\text{ }and\text{ }8\text{ }-\text{...
Find the third proportional to: (i)$2\frac{2}{3}$ (ii) a – b and a^2 – b^2
(i) take the third proportional to and 4 be x. So, , 4, x are in continued proportion. \[\begin{array}{*{35}{l}} 8/3:\text{ }4\text{ }=\text{ }4:\text{ }x \\ \left( 8/3 \right)/\text{ }4\text{...
Find the fourth proportional to: (i) 1.5, 4.5 and 3.5 (ii) 3a, 6a^2 and 2ab^2
(i) Let he fourth proportional to 1.5, 4.5 and 3.5 be x. \[\begin{array}{*{35}{l}} 1.5:\text{ }4.5\text{ }=\text{ }3.5:\text{ }x \\ 1.5~*~x\text{ }=\text{ }3.5~\times *4.5 \\ x\text{ }=\text{...
By increasing the cost of entry ticket to a fair in the ratio 10: 13, the number of visitors to the fair has decreased in the ratio 6: 5. In what ratio has the total collection increased or decreased?
Let take the cost of the entry ticket initially and at present to be 10x and 13x respectively. And let the number of visitors initially and at present be 6y and 5y respectively. Therefore,...
The bus fare between two cities is increased in the ratio 7: 9. Find the increase in the fare, if: (i) the original fare is Rs 245; (ii) the increased fare is Rs 207.
From the question we have, Increased (new) bus fare = (9/7) x original bus fare (i) Increased (new) bus fare= \[=~9/7\text{ }*\text{ }Rs\text{ }245\text{ }=\text{ }Rs\text{ }315\] Thus, the increase...
The work done by (x – 2) men in (4x + 1) days and the work done by (4x + 1) men in (2x – 3) days are in the ratio 3: 8. Find the value of x.
On assuming that the same amount of work is done one day by all the men and one day work of each man = 1 units, we get Amount of work done by (x – 2) men in (4x + 1) days = Amount of work done by (x...
The monthly pocket money of Ravi and Sanjeev are in the ratio 5: 7. Their expenditures are in the ratio 3: 5. If each saves Rs. 80 every month, find their monthly pocket money.
The pocket money of Ravi and Sanjeev are in the ratio 5: 7 Thus, we assume the pocket money of Ravi as 5k and that of Sanjeev as 7k. Also, The expenditure of Ravi and Snajeev are in the ratio 3: 5...
A school has 630 students. The ratio of the number of boys to the number of girls is 3: 2. This ratio changes to 7: 5 after the admission of 90 new students. Find the number of newly admitted boys.
Let the number of boys be 3x. Then, the number of girls = 2x \[\begin{array}{*{35}{l}} \Rightarrow \text{ }3x\text{ }+\text{ }2x\text{ }=\text{ }630 \\ 5x\text{ }=\text{ }630 \\ x\text{ }=\text{...
Divide Rs 1290 into A, B and C such that A is 2/5 of B and B: C = 4: 3.
B: C = 4: 3 so, B/C = 4/3 ⇒ C = (3/4) B And, A = (2/5) B Since, \[\begin{array}{*{35}{l}} A\text{ }+\text{ }B\text{ }+\text{ }C\text{ }=\text{ }Rs\text{ }1290 \\ \left( 2/5 \right)\text{ } B\text{...
If the ratio between 8 and 11 is the same as the ratio of 2x – y to x + 2y, find the value of 7x/ 9y.
\[\left( 2x\text{ }-\text{ }y \right)/\text{ }\left( x\text{ }+\text{ }2y \right)\text{ }=\text{ }8/11\] On cross multiplying, we get \[\begin{array}{*{35}{l}} 11\left( 2x\text{ }-\text{ }y...
Find x/y; when x^2 + 6y^2 = 5xy
Given, \[{{x}^{2}}~+\text{ }6{{y}^{2}}~=\text{ }5xy\] Dividing by y2 both side, we have Let \[\begin{array}{*{35}{l}} x/y\text{ }=\text{ }a \\ =>\text{ }{{a}^{2}}~\text{ }-5a\text{ }+\text{...
SOLVE:
SOLUTION: Since, \[\begin{array}{*{35}{l}} 3\left( m\text{ }+\text{ }n \right)\text{ }=\text{ }2\left( m\text{ }+\text{ }3n \right) \\ 3m\text{ }+\text{ }3n\text{ }=\text{ }2m\text{ }+\text{ }6n ...
What quantity must be subtracted from each term of the ratio 9: 17 to make it equal to 1: 3?
Let x be subtracted from each term of the ratio 9: 17. 27 – 3x = 17 – x 10 = 2x x = 5 Therefore, the required number that should be subtracted is 5.
Find the number which bears the same ratio to 7/33 that 8/21 does to 4/9.
Let the required number to be x/y Since, Ratio of \[8/21\text{ }to\text{ }4/9\text{ }=\text{ }\left( 8/21 \right)/\text{ }\left( 4/9 \right)\text{ }=\text{ }\left( 8/21 \right)\text{ }x\text{...
If (a – b): (a + b) = 1: 11, find the ratio (5a + 4b + 15): (5a – 4b + 3).
Since, \[\begin{array}{*{35}{l}} \left( a\text{ }\text{ }-b \right)/\text{ }\left( a\text{ }+\text{ }b \right)\text{ }=\text{ }1/\text{ }11 \\ 11a\text{ }\text{ }-11b\text{ }=\text{ }a\text{...
If a: b = 3: 8, find the value of 4a + 3b/ 6a – b.
Since, a: b = 3: 8 Therefore, a/b = 3/8
If x: y = 4: 7, find the value of (3x + 2y): (5x + y).
Since, x: y = 4: 7 Therefore, x/y = 4/7
If a: b = 5: 3, find: 5a – 3b/ 5a + 3b
Since, a: b = 5: 3 Therefore, a/b = 5/3 Now,