If the product (3)/(2)× (4)/(3)× (5)/(4)× (6)/(5)× …× (a)/(b) = 9, what is the sum of a and b?

1997 AMC 8 Problem 19 — Solution

1997 AMC 8 Problem 19

Below is the professionally curated solution for Problem 19 of the 1997 AMC 8, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1997 AMC 8 solutions, or check the answer key.

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Concepts:telescoping fraction

Difficulty rating: 1340

19.

If the product $\dfrac{3}{2}\cdot \dfrac{4}{3}\cdot \dfrac{5}{4}\cdot \dfrac{6}{5}\cdot \ldots\cdot \dfrac{a}{b} = 9,$ what is the sum of $a$ and $b?$

$11$

$13$

$17$

$35$

$37$

Solution:

Note that the numerator of each fraction cancels with the denominator of the fraction to its right.

We can then cancel out all these terms to get a final equation of $\dfrac{a}{2} = 9 \Rightarrow a = 18 \Rightarrow b = 17.$

The desired sum is then $18 + 17 = 35.$

Thus, D is the correct answer.

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