Depth · one concept from first principles · step 2 of 4
Clear denominators to reshape the condition
We know 1/a-1/b=6 but not a or b themselves. Find the value of (3a-3b-2ab)/(a-b).
Expressions
Your answer
How to solve
Strategy Convert to Algebra — Many pairs (a, b) satisfy the condition, so the answer must come from a relationship, not from values. Rewrite the condition as a-b in terms of ab, regroup the target so a-b shows, substitute, and let ab cancel.
1STEP 1
Combine the condition
Subtract the fractions over the common denominator ab.
1/a-1/b=(b-a)/(ab)=6
The condition now involves b-a and ab.
7.EE.A.2Organize Information In More Ways2STEP 2
Clear the denominator
Multiply by ab, then flip the sign to get a-b.
b-a=6ab → a-b=-6ab
a-b is just -6 times ab.
7.EE.A.2Convert To Algebrameans .
Why?
Renaming and puts both over , so the difference is .
🧱Scaling top and bottom keeps a fraction's valueMultiplying or dividing a fraction's top and bottom by the same number only renames it — the pieces change size but the amount you hold does not.
Why?
Multiplying both sides of by keeps them equal: .
🧱Do the same to both sides and equality holdsA balanced scale stays level when you take the same amount off both pans or add the same amount to both.
Why?
is times , so multiplying both sides by flips each to the other side of zero: .
🧱Signs of products and quotientsMultiplying by a negative number flips a number to the other side of zero, so two flips land back on the positive side: for two nonzero numbers, a product or quotient with like signs is positive and one with unlike signs is negative (zero itself has no sign).
3STEP 3
Let a-b show in the top
Factor 3 out of the first two terms of the numerator.
3a-3b-2ab=3(a-b)-2ab
Now both top and bottom are written with a-b.
6.EE.A.3Identify Subproblems4STEP 4
Substitute and cancel
Replace a-b with -6ab everywhere; ab cancels because it is not 0.
(3(-6ab)-2ab)/(-6ab)=-20ab/(-6ab)=10/3
Only a number is left.
7.EE.A.2Convert To AlgebraAnswer
10/3
Take a=1/7, b=1: 7-1=6 works. Then 3a-3b-2ab=3/7-3-2/7=-20/7 and a-b=-6/7, so the value is 20/6=10/3.
Takeaway
When the letters can't be pinned down, reshape the condition until it fits the expression you need.
- Combine the condition
- Clear the denominator
- Let a-b show in the top
- Substitute and cancel
Where next?
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