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Multi-Step Equations with Variables on Both Sides, a la Shmoop.

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Runny the Reindeer has always tried to follow in the hoofsteps of his brother Rudolph. [Reindeer walking]

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Every year, Rudolph buys Santa an expensive new hat. [Rudolph giving Santa a new hat]

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Out of the goodness of his heart, Rudolph gives Runny exactly half his cash, so they’ll [Rudolph giving the cash to Runny]

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have the same amount to spend on Christmas gifts.

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On the way to the Sugarplum Mall, a cute vixen recognizes Rudolph and offers to give him [Vixen approaching Rudolph]

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7 dollars to take a photo with him and his famous nose.

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Rudolph doesn’t want Runny to feel left out…so he pays the vixen 2 dollars to take [Rudolph pushes Runny to the front]

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a photo with Runny and his…slightly less famous shnoz.

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Rudolph even asks the vixen to give the $7 to Runny instead. [The vixen gives the 7 dollars over]

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At checkout, Runny has exactly enough money to purchase two hats, while Rudolph has exactly [Runny handing over 2 hats at the checkout]

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enough to buy one.

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Can you determine the price of one of these hats?

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Oof.

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That’s a lot of information.

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It will help to be able to solve this sort of problem, however, because it’s the type [Rudolph's nose glowing red]

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of situation that pops up a lot in real life. [Girl counting money]

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First, we need to determine the variable we’re going to use.

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We don’t know how much money each reindeer started with, but we do know that they started [Both reindeers holding wallets]

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with the same amount…

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…since the problem tells us that Rudolph gave half his cash to Runny before they headed

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out.

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So let’s call their starting cash “x.”

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Same for both.

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Rudolph gave two dollars to the cute vixen. [Rudolph handing the money over]

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So…x minus 2.

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Runny, on the other hand, received seven dollars from his fangirl, so…x plus 7.

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However, we’re also told that Runny is able to buy twice as many hats as Rudolph after

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this little exchange…

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…which means that we’d have to multiply the expression for Rudolph by 2 to set these [Both reindeer holdingh their hats]

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equal to one another.

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So…

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2 times the quanitity x minus 2…equals x plus 7.

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Multiplying out the 2 on the left-hand side, we get 2x minus 4 equals x plus 7.

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Now, of course, we want to get x by itself so we know how much cashola these guys started

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with.

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So we can subtract x from both sides to get x minus 4 equals 7…

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…then add 4 to each side to get x equals 11. [The equation workings]

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And…there we have it.

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Each reindeer started their shopping trip with $11. [Reindeer with their wallets in front of a Christmas tree]

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But…that’s not what the question is asking.

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We know that Rudolph gave away $2, and still had exactly enough money to buy 1 hat.

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So if he started with $11, and gave away $2, leaving himself with $9…then each hat must

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run $9 on the button. [Christmas hat with a pricetag]

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We can double-check to see if that’s correct by plugging the same number into Runny’s

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equation.

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He started with $11, and the vixen gave him another $7, giving him $18 total…

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…and sure enough, 18 divided by 9 is 2, meaning Runny would have had exactly enough

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cash to buy 2 hats.

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Looks like Santa has a new favorite this year. [Santa stood with presents by a Christmas tree]

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At least until allergy season strikes… [Runny sneezes and blows Santa over]

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