Initial Publication Date: April 12, 2016
Rearranging Equations: Sample Problems
Problem 1:
You run a pizza parlor and learn that an economic study shows:
the price elasticity of demand for pizza is -1.5
Use this information to predict what will happen if you raise the price of pizza by 25%.
Elasticity of demand can be represented by the following equation:
Elasticity = % change in Q / % change in P
Or, written another way:
`"Elasticity" = (\%\DeltaQ)/(\%\DeltaP)`
To solve this problem, rearrange the above equation to isolate the % change in Q, or %ΔQ.
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Start with the equation for elasticity and plug in -1.5 for elasticity and 25 for the % change in P, or %ΔP.
$-1.5 = (\%\Delta Q) / 25$
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Multiply both sides by 25 and simplify.
$(-1.5) \times 25 = ((\%\Delta Q) / 25) \times 25$
$-37.5 = ((\%\Delta Q) /25) \times 25$
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Cancel out the 25's on the right hand side.
$\require{cancel}-37.5 = ((\%\Delta Q) / \cancel{25}) \times \cancel{25}$
$\%\Delta Q = -37.5$
When the price of pizza increases by 25%, the quantity demand will fall by 37.5%.
Problem 2:
You read economic data for a country showing that:
- current nominal GDP is 180,350 million
- real GDP is 175,100 million
If you can find the GDP deflator, you'll learn about inflation in that country. Solve for the GDP deflator.
Real GDP is calculated with the following equation:
`"Real GDP" = ("Nominal GDP"/"GDP Deflator") xx 100`
We can rearrange this equation to produce a new equation for the GDP deflator.
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Start with the equation for real GDP and plug in 180,350 for nominal GDP and 175,100 for real GDP.
$175{,}100 = ((180{,}350) / \text{GDP Deflator}) \times 100$
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Divide both sides by 100.
$175{,}100 / 100 = ((180{,}350 / \text{GDP Deflator}) \times 100) / 100$
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Cancel out the 100's on the right hand side and simplify.
$175{,}100 / 100 = (( 180{,}350 / \text{GDP Deflator} ) \times \cancel{100 ) / 100}$
$1{,}751 = 180{,}350 / \text{GDP Deflator}$
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Multiply both sides by the GDP Deflator.
$1{,}751 \times \text{GDP Deflator} = ( 180{,}350 / \text{GDP Deflator} ) \times \text{GDP Deflator}$
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Cancel out the GDP Deflator's on the right hand side.
$1{,}751 \times \text{GDP Deflator} = (180{,}350 / \cancel{ \text{GDP Deflator} ) \times \text{GDP Deflator}}$
$1{,}751 \times \text{GDP Deflator} = 180{,}350$
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Divide both sides by 1,751.
$(1{,}751 \times \text{GDP Deflator}) / 1{,}751 = 180{,}350 / 1{,}751$
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Cancel out the 1,751's on the left hand side and simplify.
$(\cancel{1{,}751} \times \text{GDP Deflator}) / \cancel{1{,}751} = 180{,}350 / 1{,}751$
$\text{GDP Deflator} = 102.998$
The GDP deflator is 102.998.
(Note: that means since the base year, prices went up 2.998%)
Problem 3:
A consumer's budget constraint is represented by the equation:
`100 = 4Y + 2X`
in which Y is the quantity of health care, X is the quantity of food.
An economist wants to understand a trade-off: for each dollar spent on food, how much less is available for health care. In other words, rearrange the equation for Y.
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Subtract 2X from both sides.
$100 - 2X = 4Y + 2X - 2X$
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Cancel out the 2X's on the right hand side.
$100 - 2X = 4Y + \cancel{2X - 2X}$
$100 - 2X = 4Y$
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Divide both sides by 4.
$(100 - 2X)/4 = 4Y/4$
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Cancel out the 4's on the right hand side and simplify.
$(100 - 2X)/4 = \cancel{4}Y/\cancel{4}$
$Y = (100 - 2X)/4$
$Y = 100/4 - 2X/4$
$Y = 25 - X/2$
Y = 25 - X/2
Problem 4:
An economist knows how much total is produced in a country (GDP) and how much is purchased domestically (C, I and G). With the information, how much was lost or gained to the difference between exports and imports, or net exports (NX).
Rearrange GDP = C + I + G + NX to solve for NX.
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Subtract C, I and G from both sides.
$GDP - C - I - G = C + I + G + NX - C - I - G$
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Cancel out the C's, I's and G's on the right hand side.
$GDP - C - I - G = \cancel{C + I + G} + NX \cancel{- C - I - G}$
$GDP - C - I - G = NX$
NX = GDP - C - I - G
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