Many word problems seem to require us to write formulas in order to solve. Certain problems, though, qualify for a neat technique: Smart Numbers. We can actually pick our own real numbers and use them to solve!
Set your timer for 2 minutes for this Fill-In problem and GO! (© ManhattanPrep)
* Lisa spends 3/8 of her monthly paycheck on rent and 5/12 on food. Her roommate, Carrie, who earns twice as much as Lisa, spends ¼ of her monthly paycheck on rent and ½ on food. If the two women decide to donate the remainder of their money to charity each month, what fraction of their combined monthly income will they donate? (Assume all income in question is after taxes.)
(No answer choices given; this is a fill-in-the-blank)
We’ve got two women, Lisa and Carrie, and they each spend a certain proportion of income on rent and on food. Annoyingly, the fractions don’t have the same denominators; even more annoyingly, the two women don’t make the same amount of money. All of that will make an algebraic solution challenging.
Here’s what an algebraic solution would look like. Let’s call Lisa’s income x. She spends (3/8)x on rent and (5/12)x on food. Add these together:
(3x/8) + (5x/12) = (9x/24) + (10x/24) = 19x/24
Subtract from 100%, or x:
24x/24 “ 19x/24 = 5x/24
Lisa donates 5/24 of x, her income, to charity. What about Carrie?
Carrie’s income is equal to 2x (because she makes twice as much as Lisa). How much does she spend on rent and food?
(1/4)(2x) + (1/2)(2x) = (1x/2) + (2x/2) = 3x/2
This looks very strange, because it seems like she spends more than she has “ 3/2 is equal to 1.5, which is greater than 1. Remember, though, that she started out earning 2x, not 1x. Subtract from 100%, or 2x, to get the remaining money:
2x “ 3x/2 = 4x/2 “ 3x/2 = x/2
Carrie donates an amount equivalent to ½ of Lisa’s income (x). Together, they donate how much?
(5/24)x + (1/2)x = (5/24)x + (12/24)x = (17/24)x
And now here’s the final tricky step. That’s not our answer! 17/24 represents the total amount they donate as a percentage of Lisa’s income alone. (Remember that we calculated all along using x, which represents Lisa’s income.) The problem asks us to find the amount donated as a percentage of their combined incomes. Lisa’s income is x and Carrie’s income is 2x, so their combined income is 3x, or three times Lisa’s income alone. If 17/24 represents the percentage based on Lisa’s income, we have to multiply the denominator by 3 to represent the total income: 17/(24´3) = 17/72.
Seriously? There must be an easier way!
There are multiple better ways, in fact, including better algebra methods (though the above is one of the most common methods people try to use, though they typically make mistakes along the way). If algebra is your strength, then possibly you’ll immediately see one of the better algebra methods and use that.
But there’s an even better method than algebra: arithmetic, or real numbers. Everyone on the planet has been doing arithmetic longer, and everyone on the planet is more comfortable with arithmetic than with algebra, no matter how good at algebra you are. If I still haven’t convinced you, I’ll say one more thing: I’m really good at algebra, but I’d still rather pick a real number on a problem like this. : )
Nowhere in the problem are we given a real value or a real amount of money for either Lisa or Carrie “ we’re only given fractions of total income. The question also asks for a fractional amount of their income. These clues tell us that the problem qualifies for the Smart Numbers technique: we pick our own real number (or numbers) to use, thereby avoiding algebra entirely.
Let’s start with the person who’s making the smaller amount of money: Lisa. The fraction for her rent has a denominator of 8, while the fraction for her food has a denominator of 12. Ideally, we want to pick the smallest number that will work well with both of these numbers. The least common multiple of 8 and 12 is 24, so 24 is a good number to pick. (Note: when the starting numbers are small, the easiest way to find the LCM is simply to list out numbers until you find a match. 8, 16, 24. 12, 24.)
So let’s say that Lisa makes $24 per month. (It doesn’t matter that this probably isn’t what someone would really make in the real world. We only care about picking a number that will make the problem easy!)
If Lisa makes $24, then she spends 3/8, or (24)(3/8) = $9 on rent, and she spends (24)(5/12) = $10 on food. The amount left over is 24 “ 9 “ 10 = $5.
If Lisa makes $24, then Carrie must make $48, because the problem told us that Carrie earns twice as much as Lisa. Carrie spends (48)(1/4) = $12 on rent, and (48)(1/2) = $24 on food, leaving her with 48 “ 12 “ 24 = $12 left over.
Together, the two women donate a total of 5 + 12 = $17 to charity. What fraction does this represent of their total monthly income? Their combined income is 24 + 48 = $72, so the women donate 17/72 of their income to charity.
The correct answer is 17/72.
Did you make any mistakes along the way? One common incorrect answer is 17/48; for this answer, someone correctly calculated the 17 portion, but mistakenly said that the two women each earned $24 per month, forgetting that Carrie makes twice as much. Another common error results in the answer 17/24: forgetting to add Carrie’s income to the total. This is the answer you’ll get if you follow the algebra path but forget that very last step (multiplying the denominator by 3).
Two other common incorrect answers are 11/24 or 11/48. If we (incorrectly) use the same monthly income for both women throughout the problem, we’ll determine that the two women donate $11 rather than $17 to charity. (The two possible denominators, 24 and 48, represent the errors we discussed in the previous paragraph.)
Key Takeaways for Word Problems without Real Numbers
(1) When a problem keeps talking about something “ an income, the price of a TV, whatever it might be! “ but never provides real numbers for that thing, just fractions, percentages, variables, or similar, you can pick your own real numbers.
(2) Smart numbers are ones that will work nicely in the problem. For fraction problems, use common denominators. For percentage problems, 100%, 50% and 20% can work well. For straight algebra problems (with variables), small integers such as 2, 3, and 5 work well (avoid 0 and 1).
(3) Practice converting these kinds of problems enough times that you feel comfortable picking your own real numbers. Once you’ve converted the problem to arithmetic, you’ll be better off, since you’re already better at working with real numbers (arithmetic) than you are with algebra.
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