Depth · one concept from first principles · step 2 of 4
Sum of reciprocals is sum over product
Three unknown numbers have pairwise sums a, b, c and product 1. Express 1/x{y}+1/y{z}+1/z{x} in terms of a, b, c.
Expressions
Your answer
How to solve
Strategy Convert to Algebra — Name the three numbers. Instead of finding each one, look at what the target needs: over a common denominator the reciprocal sum is just (sum of the numbers) divided by (their product), and both of those come straight from the givens.
1STEP 1
Name the numbers
Call the numbers x, y, z and write the givens as equations.
x+y=a, y+z=b, z+x=c, xyz=1
Four facts about three letters.
7.EE.B.4Convert To Algebra2STEP 2
Add the three sums
Adding the three pairwise sums counts every number twice.
2(x+y+z)=a+b+c
So the total of the three numbers is half of a+b+c.
7.EE.A.2Change Focus Count The Complement3STEP 3
Common denominator
Give the three reciprocals the common denominator xyz: each missing letter moves to the top.
1/x{y}+1/y{z}+1/z{x}=(z+x+y)/xyz
The top becomes the sum, the bottom the product.
7.EE.A.2Change Focus Count The Complement.
Why?
Multiplying top and bottom of by gives , and likewise the others become and .
🧱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?
Fractions cut into the same pieces add by adding their tops.
🧱A fraction is equal shares of one wholeThe bottom number names how many equal pieces one whole was cut into and the top counts them, which is why same-sized pieces add just by counting.
4STEP 4
Substitute
Put in x+y+z=(a+b+c)/2 and xyz=1.
(x+y+z)/(xyz)=(a+b+c)/2 ÷ 1=(a+b+c)/2
No single number was ever needed.
6.EE.A.3Identify SubproblemsAnswer
(a+b+c)/2
Try x=y=z=1: the product is 1, each pairwise sum is 2, and the reciprocal sum is 1+1+1=3. The formula gives (2+2+2)/2=3.
Takeaway
Before solving for each unknown, check whether the question only needs their sum and product.
- Name the numbers
- Add the three sums
- Common denominator
- Substitute
Where next?
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