RATIO & PROPORTION (20/21)
P1
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
3) {
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
2).
