Thinking Mathematically (6th Edition)

Published by Pearson
ISBN 10: 0321867327
ISBN 13: 978-0-32186-732-2

Chapter 5 - Number Theory and the Real Number System - 5.3 The Rational Numbers - Exercise Set 5.3: 123

Answer

$\frac{1}{3}$ cup of butter, 2.5 ounces of unsweetened chocolate, $\frac{3}{4}$ cup of sugar, 1 teaspoon of vanilla, 1 egg, and $\frac{1}{2}$ cup of flour are needed to make 8 brownies.

Work Step by Step

The original recipe makes 16 brownies. To adjust it to make 8 brownies, we adjust each ingredient by a factor of $\frac{8}{16}$ which reduces to $\frac{1}{2}$. We do this because 8 is one-half of sixteen. Note: all answers in fraction form are reduced to lowest terms for the final answers. For 16 brownies, we need $\frac{2}{3}$ cup of butter. So, for 8 brownies, we need $\frac{2}{3}$ x $\frac{1}{2}$ = $\frac{2}{6}$ = $\frac{1}{3}$ cup of butter. For 16 brownies, we need 5 ounces of unsweetened chocolate. To make 8 brownies, we need just half of that: $\frac{5}{1}$ x $\frac{1}{2}$ = $\frac{5}{2}$ ounces of unsweetened chocolate. In decimal form, this is 2.5 ounces of unsweetened chocolate. For 16 brownies, we need 1$\frac{1}{2}$ cup of sugar. Before we calculate for 8 brownies, let’s change 1$\frac{1}{2}$ to an improper fraction so that we can properly multiply two fractions. The whole number 1 can be viewed as 2 halves. So 1$\frac{1}{2}$ = $\frac{3}{2}$. Therefore, to make 8 brownies, we need $\frac{3}{2}$ x $\frac{1}{2}$ = $\frac{3}{4}$ cups of sugar. For 16 brownies, we need 1 teaspoon of vanilla. Half of 1 is one-half, so we need $\frac{1}{2}$ teaspoon of vanilla to make 8 brownies. For 16 brownies, we need 2 eggs. For 8 brownies we need one-half the amount. Half of 2 is 1. Therefore, we need 1 egg to make 8 brownies. For 16 brownies, we need 1 cup of flour. For 8 brownies we need one-half of this amount or $\frac{1}{2}$ cup of flour.
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