I tried to multiply the equation by a then b then c... I don't know please help by showing the steps I am very lost
Determine a+b+c
25/84=1/a+1/an+1/abc
Sorry it was supposed to be
Determine a+b+c
25/84=1/a+1/ab+1/abc
It's hard to explain.
Because of that in google type:
CRUX MATHEMATICORUM Contest Corner Problem CC210
When you see list of results click on:
There are exactly two positive integer triples (a , b , c) such that , 25/84 ...
and read answer.
To determine the value of a+b+c in the equation 25/84 = 1/a + 1/an + 1/abc, we need to simplify and solve the equation. Here's how you can approach it step by step:
Step 1: Find the common denominator
To add fractions, we need to find a common denominator. In this case, the denominator for the three fractions is "abc". So, multiply each fraction by the appropriate factor to achieve the common denominator.
25/84 = 1/a + 1/an + 1/abc
Multiply the first fraction by bc:
25/84 = bc/abc + 1/an + 1/abc
Step 2: Combine the fractions
Since we now have a common denominator, we can combine the fractions on the right side of the equation:
25/84 = (bc + 1/an + 1)/abc
Step 3: Cross-multiply
To eliminate the fraction on the right side, cross-multiply:
25 * abc = 84 * (bc + 1/an + 1)
25abc = 84(bc + 1/an + 1)
Step 4: Distribute and simplify
Distribute 84 to the terms inside the parentheses:
25abc = 84bc + 84/an + 84
Step 5: Simplify further
To simplify, let's rewrite 84/an as 84n/an:
25abc = 84bc + 84n/an + 84
Step 6: Rearrange the equation
Move the terms containing "n" to the left side and the others to the right side:
25abc - 84n/an = 84bc + 84
Step 7: Rearrange further
To eliminate the fraction, multiply both sides by "an":
25abc * an - 84n = 84bc * an + 84 * an
Step 8: Simplify even more
On the left side, simplify by canceling out "an":
25abcn - 84n = 84abcn + 84n
Step 9: Combine like terms
Combine the terms with "n" together:
25abcn - 84n - 84abcn - 84n = 0
Step 10: Simplify one last time
Combine the terms on the left side:
(25abcn - 84abcn) - (84n + 84n) = 0
(25abc - 84abc) n - (84 + 84)n = 0
-59abcn - 168n = 0
Step 11: Factor out "n"
Factor out "n":
n(-59abc - 168) = 0
Step 12: Solve for "n"
Since the product of two factors equals zero, set each factor equal to zero:
n = 0
-59abc - 168 = 0
Step 13: Solve for "abc"
For the second equation, rearrange it to solve for "abc":
-59abc = 168
abc = 168 / -59
At this point, we have solved for the values of "n" and "abc". However, the original question asks for the value of a+b+c, not "abc".