Depth · one concept from first principles · step 2 of 4

Rewrite numerators using the given total

The constants $a$ , $b$ , $c$ satisfy $3a+2b+c=20$ , with $abc\neq 0$ and $\frac{1}{3a}+\frac{1}{2b}+\frac{1}{c}\neq 0$ . Solve the equation $\frac{x}{3a}+\frac{x}{2b}+\frac{x}{c}-6=\frac{4b+2c}{3a}+\frac{6a+2c}{2b}+\frac{6a+4b}{c}$ .

Expressions

Your answer

How to solve

Strategy Change Focus / Count the Complement — Instead of clearing denominators, look at each numerator: 4b+2c is twice 2b+c, which is the total 20 minus 3a. Rewriting all three this way makes the right side look exactly like the left side, with 40 in place of x.

1STEP 1

Rewrite the first numerator

2b+c=20-3a, so double it.

4b+2c=2(20-3a)=40-6a

2STEP 2

Rewrite the other two

In the same way, 3a+c=20-2b and 3a+2b=20-c.

6a+2c=40-4b, 6a+4b=40-2c

3STEP 3

Split each fraction

Each fraction becomes 40 over its denominator minus 2.

(40-6a)/(3a)+(40-4b)/(2b)+(40-2c)/c=40/(3a)+40/(2b)+40/c-6

4STEP 4

Compare the two sides

Add 6 to both sides and factor: x(1/(3a)+1/(2b)+1/c)=40(1/(3a)+1/(2b)+1/c). The bracket is not 0, so divide by it.

x=40

Answer

x=40

Try particular constants: a=2, b=3, c=8 satisfy 6+6+8=20. The left side at x=40 is 40/6+40/6+40/8-6=40/3+5-6=37/3. The right side is 28/6+28/6+24/8=28/3+3=37/3. Equal.

Takeaway

When a total is given, look for pieces that are "the total minus something" and rewrite them.

Rewrite the first numerator
Rewrite the other two
Split each fraction
Compare the two sides

Where next?

Level upsuggested

Clear denominators to reshape the condition Sum of reciprocals is sum over product

Another one like thissuggested

12 practice problems

Same look, different logicsuggested

Each wrong solve still reveals one true constant Write the solution in a, then filter by wholeness

Step back and solidifysuggested

Equal side products cancel the shared corner A relation fixes a ratio, so the expression is constant

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▶ Practice — 12 problems