Unknown Renowned Enry

Algebra Level 3

Calvin bought himself three new cool math books from renowned authors Henry, Renry and Fenry.

However, he lost his receipts, so he doesn't know how much each one cost. He remembers, though, that Henry's book was 75% to 80% the price of Renry's, and 30% to 40% the price of Fenry's.

We can deduce that Renry's book was a a % to b b % the price of Fenry's. What is a ⋅ b a \cdot b ?

1000 4000 3000 2000

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Guilherme Dela Corte by Guilherme Dela Corte , 24, Brazil

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1 solution

Let H , R , F H, R, F respectively be the prices of Henry's, Renry's and Fenry's math books.

We know that H = ( 75 % → 80 % ) R = ( 30 % → 40 % ) F H = (75 \% \to 80 \%)R = (30 \% \to 40 \%) F . Thus R = 30 % → 40 % 75 % → 80 % F R = \dfrac{30 \% \to 40 \%}{75 \% \to 80 \%} F .

To minimize the ratio R F \dfrac{R}{F} , we must maximize the denominator (80%) and minimize the numerator (30%). This yields R = 37.5 % F R = 37.5 \% F .

To maximize the ratio R F \dfrac{R}{F} , we must maximize the numerator (40%) and minimize the denominator (75%). This yields R = 53. 3 ˉ % F R = 53.\bar{3} \% F .

Finally, a b = 37.5 ⋅ 53. 3 ˉ = 2000. ab = 37.5 \cdot 53.\bar{3} = \boxed{2000.}

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