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In 2021 a university’s total operating expenses rose by 6%, where the expenses consist of administrative expenses and research expenses. If administrative expenses rose by 5% and research expenses rose by 9% during 2021, then research expenses represented what percent of operating expenses at the beginning of 2021?

Explanation	Calculations	Help
$We first create variables for the beginning-of-year expenses$	$A = administrative expenses at beginning of 2021 R = research expenses at beginning of 2021$	Theory & Tactics Method Card TRANS1-A Click to view full details
$We translate the percent increases for each component "administrative expenses rose 5 percent" and "research expenses rose 9 percent"$	$1.05 A = A + 0.05 A 1.09 R = R + 0.09 R$	Theory & Tactics Method Card TRANS2-D Click to view full details
$We translate the overall percent increase "total operating expenses rose 6 percent"$	$1.06 (A + R) = (A + R) + 0.06 (A + R)$	Theory & Tactics Method Card TRANS2-D Click to view full details
$We set the new total expenses from components equal to the increased total expenses$	$1.05 A + 1.09 R = 1.06 (A + R)$	Theory & Tactics Method Card TRANS1-B Click to view full details
$We expand and collect like terms to find a relationship between R and A$	$1.05 A + 1.09 R = 1.06 A + 1.06 R 1.05 A - 1.06 A + 1.09 R - 1.06 R = 0 - 0.01 A + 0.03 R = 0 0.03 R = 0.01 A},{$
$We find the fraction of total operating expenses that research expenses represented at the beginning of 2021$	$\frac{R}{A} = \frac{0.01}{0.03} = \frac{1}{3} \frac{R}{A + R} = \frac{\frac{1}{3}}{1 + \frac{1}{3}} = \frac{\frac{1}{3}}{\frac{4}{3}} = \frac{1}{4} = 0.25 = 25%$	Theory & Tactics Method Card FDPR2-A Click to view full details

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C

In 2021 a university’s total operating expenses rose by 6%, where the expenses consist of administrative expenses and research expenses. If administrative expenses rose by 5% and research expenses rose by 9% during 2021, then research expenses represented what percent of operating expenses at the beginning of 2021?

Explanation	Calculations	Help
$We first create variables for the beginning-of-year expenses$	$A = administrative expenses at beginning of 2021 R = research expenses at beginning of 2021$	Theory & Tactics Method Card TRANS1-A Click to view full details
$We translate the percent increases for each component "administrative expenses rose 5 percent" and "research expenses rose 9 percent"$	$1.05 A = A + 0.05 A 1.09 R = R + 0.09 R$	Theory & Tactics Method Card TRANS2-D Click to view full details
$We translate the overall percent increase "total operating expenses rose 6 percent"$	$1.06 (A + R) = (A + R) + 0.06 (A + R)$	Theory & Tactics Method Card TRANS2-D Click to view full details
$We set the new total expenses from components equal to the increased total expenses$	$1.05 A + 1.09 R = 1.06 (A + R)$	Theory & Tactics Method Card TRANS1-B Click to view full details
$We expand and collect like terms to find a relationship between R and A$	$1.05 A + 1.09 R = 1.06 A + 1.06 R 1.05 A - 1.06 A + 1.09 R - 1.06 R = 0 - 0.01 A + 0.03 R = 0 0.03 R = 0.01 A},{$
$We find the fraction of total operating expenses that research expenses represented at the beginning of 2021$	$\frac{R}{A} = \frac{0.01}{0.03} = \frac{1}{3} \frac{R}{A + R} = \frac{\frac{1}{3}}{1 + \frac{1}{3}} = \frac{\frac{1}{3}}{\frac{4}{3}} = \frac{1}{4} = 0.25 = 25%$	Theory & Tactics Method Card FDPR2-A Click to view full details

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C