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Solution
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Solution provided by AtoZmath.com
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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
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1. If `a:b:c=2:3:5` then find value of `(a^2+b^2+c^2)/(ab+bc+ca)`
 1. If `a/b=2/3`, then find `(2a-3b)/(a-2b)`
2. If `a:2=b:3=c:5`, then find `(a^2+b^2+c^2)/(ab+bc+ca)`
3. If `a/4=b/15`, then find `(a+b)/(a-b)`
4. If `(2a+3b)/(a+b)=13/5`, then find `a/b`
5. If `(x^2+1)/(x^2-1)=2`, then find `x`
6. If `x:30=12:20`, then find `x`
2. If `a:b=2:3,b:c=4:5` then find `a:b:c`
1. If `a:b=2:3,b:c=4:5`, then find `a:b:c`
2. If `b:c=4:5,c:d=5:6`, then find `b:c:d`
3. If `a:b=2:3,b:c=4:5,c:d=5:6`, then find `a:b:c:d`
4. If `2a=3b`, then find `a:b`
5. If `2a=3b=7c`, then find `a:b:c`
3. If `a/b=c/d=e/f` then prove that `(2a+3c-4e)/(2b+3d-4f)=(5a-4c+3e)/(5b-4d+3f)`
1. If `a/b=c/d=e/f`, then prove that `(2a+3c-4e)/(2b+3d-4f)=(5a-4c+3e)/(5b-4d+3f)`
2. If `x/(b^2-c^2)=y/(c^2-a^2)=z/(a^2-b^2)`, then prove that `x+y+z=0`
4. If `x/(y+z)=y/(z+x)=z/(x+y)` then prove the value of each ratio is `1/2` or `-1`
1. If `x/(y+z)=y/(z+x)=z/(x+y)`, then prove each ratio = `1/2,-1`
2. If `(5a+6b)/(7c)=(6b+7c)/(5a)=(7c+5a)/(6b)`, then prove each ratio = `2,-1`
3. If `(ax+by)/(x+y)=(ay+bz)/(y+z)=(az+bx)/(z+x)`, then prove each ratio = `(a+b)/2`
5. Geometric Mean
1. Geometric Mean of `49/4` and `16/9`
2. Geometric Mean of `8a^2` and `32b^2`
3. Geometric Mean of `(a-b)^2` and `(a+b)^2`
4. If geometric mean `6x^2` and x is `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
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Example1. If `x/(y+z)=y/(z+x)=z/(x+y)` then prove the value of each ratio is `1/2,-1`
Solution:
Here `x/(y+z)=y/(z+x)=z/(x+y)`
Case-1 : If `x+y+z!=0`, then
Each ratio`=(x+y+z)/(y+z+z+x+x+y)`
`=(x+y+z)/(2y+2z+2x)`
`=(x+y+z)/(2(y+z+x))`
Cancel the common factor `(x+y+z)`
`=(1)/(2)`
Case-2 : If `x+y+z=0`, then
`y+z=-x`
Then, the first ratio `=(x)/(y+z)`
`=(x)/(-x)`
Cancel the common factor `-x`
`=-1`
Hence, each ratio `=-1`.
Thus, the value of each ratio is `(1)/(2)` or `-1`.
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