Ratio and Proportion - If x/(y+z)=y/(z+x)=z/(x+y) then prove the value of each ratio is 1/2 or -1 calculator

SolutionHelp

1. If a:b:c=2:3:5 $a:b:c=2:3:5$ then find value of a2+b2+c2ab+bc+ca $(a^2+b^2+c^2)/(ab+bc+ca)$

1. If

ab=23 $a/b=2/3$ , then find

2a-3ba-2b $(2a-3b)/(a-2b)$

2. If

a:2=b:3=c:5 $a:2=b:3=c:5$ , then find

a2+b2+c2ab+bc+ca $(a^2+b^2+c^2)/(ab+bc+ca)$

3. If

a4=b15 $a/4=b/15$ , then find

a+ba-b $(a+b)/(a-b)$

4. If

2a+3ba+b=135 $(2a+3b)/(a+b)=13/5$ , then find

ab $a/b$

5. If

x2+1x2-1=2 $(x^2+1)/(x^2-1)=2$ , then find

x $x$

6. If

x:30=12:20 $x:30=12:20$ , then find

x $x$

2. If a:b=2:3,b:c=4:5 $a:b=2:3,b:c=4:5$ then find a:b:c $a:b:c$

1. If

a:b=2:3,b:c=4:5 $a:b=2:3,b:c=4:5$ , then find

a:b:c $a:b:c$

2. If

b:c=4:5,c:d=5:6 $b:c=4:5,c:d=5:6$ , then find

b:c:d $b:c:d$

3. If

a:b=2:3,b:c=4:5,c:d=5:6 $a:b=2:3,b:c=4:5,c:d=5:6$ , then find

a:b:c:d $a:b:c:d$

4. If

2a=3b $2a=3b$ , then find

a:b $a:b$

5. If

2a=3b=7c $2a=3b=7c$ , then find

a:b:c $a:b:c$

3. If ab=cd=ef $a/b=c/d=e/f$ then prove that 2a+3c-4e2b+3d-4f=5a-4c+3e5b-4d+3f $(2a+3c-4e)/(2b+3d-4f)=(5a-4c+3e)/(5b-4d+3f)$

1. If

ab=cd=ef $a/b=c/d=e/f$ , then prove that

2a+3c-4e2b+3d-4f=5a-4c+3e5b-4d+3f $(2a+3c-4e)/(2b+3d-4f)=(5a-4c+3e)/(5b-4d+3f)$

2. If

xb2-c2=yc2-a2=za2-b2 $x/(b^2-c^2)=y/(c^2-a^2)=z/(a^2-b^2)$ , then prove that

x+y+z=0 $x+y+z=0$

4. If xy+z=yz+x=zx+y $x/(y+z)=y/(z+x)=z/(x+y)$ then prove the value of each ratio is 12 $1/2$ or -1 $-1$

1. If

xy+z=yz+x=zx+y $x/(y+z)=y/(z+x)=z/(x+y)$ , then prove each ratio =

12,-1 $1/2,-1$

2. If

5a+6b7c=6b+7c5a=7c+5a6b $(5a+6b)/(7c)=(6b+7c)/(5a)=(7c+5a)/(6b)$ , then prove each ratio =

2,-1 $2,-1$

3. If

ax+byx+y=ay+bzy+z=az+bxz+x $(ax+by)/(x+y)=(ay+bz)/(y+z)=(az+bx)/(z+x)$ , then prove each ratio =

a+b2 $(a+b)/2$

5. Geometric Mean

1. Geometric Mean of

494 $49/4$ and

169 $16/9$

2. Geometric Mean of

8a2 $8a^2$ and

32b2 $32b^2$

3. Geometric Mean of

(a-b)2 $(a-b)^2$ and

(a+b)2 $(a+b)^2$

4. If geometric mean

6x2 $6x^2$ and x is

9x $9x$ , find value of x

6. Ratios(duplicate, triplicate) and proportional(mean, third, fourth)

1. Duplicate ratio of 2:3

2. Triplicate ratio of 2:3

3. Sub-Duplicate ratio of 16:25

4. Sub-Triplicate ratio of 8:27

5. Compounded ratio of 2:5,3:4

6. Compounded ratio of 2:3,3:4,4:5

7. Mean proportional of 8,32

8. Third proportional of 16,36

9. Fourth proportional of 3,7,15

10. Compare ratios of 2:3,3:4,4:5

Example

1. If xy+z=yz+x=zx+y $x/(y+z)=y/(z+x)=z/(x+y)$ then prove the value of each ratio is 12,-1 $1/2,-1$

Solution:
Here

xy+z=yz+x=zx+y $x/(y+z)=y/(z+x)=z/(x+y)$

Case-1 : If

x+y+z≠0 $x+y+z!=0$ , then

Each ratio

=x+y+zy+z+z+x+x+y $=(x+y+z)/(y+z+z+x+x+y)$

=x+y+z2y+2z+2x $=(x+y+z)/(2y+2z+2x)$

=x+y+z2(y+z+x) $=(x+y+z)/(2(y+z+x))$

Cancel the common factor

(x+y+z) $(x+y+z)$

=12 $=(1)/(2)$

Case-2 : If

x+y+z=0 $x+y+z=0$ , then

y+z=-x $y+z=-x$

Then, the first ratio

=xy+z $=(x)/(y+z)$

=x-x $=(x)/(-x)$

Cancel the common factor

-x $-x$

=-1 $=-1$

Hence, each ratio

=-1 $=-1$ .

Thus, the value of each ratio is

12 $(1)/(2)$ or

-1 $-1$ .