SAT Mathematics : Translating Words to Inequalities

Study concepts, example questions & explanations for SAT Mathematics

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Example Questions

Example Question #1 : Translating Words To Inequalities

On a trip to visit a friend at college, Hayley is careful to never exceed the speed limit of 70 miles per hour. Which of the following represents the time in hours, , that it takes her to drive the  miles between two highway exits during her trip?

Possible Answers:

Correct answer:

Explanation:

For this example, you need to carefully analyze the well-known equation Distance = Rate x Time. The problem tells you that Hayley will not exceed 70 miles per hour which means her speed is always ≤ 70 mph. Since the problem asks you to isolate the limit for time on the trip, you should manipulate the previous formula to become: 

Time is given as the variable h, for hours, and distance is given as the variable m, for miles. If you were told that Hayley drove 70 mph for the entire trip then the correct equation would be: h = 70 . The tricky part of this problem, however, is that you must get the inequality sign correct given that her speed never exceeds 70mph.

Remember that the slower you drive, the greater the time becomes. Since an entire trip at 70mph would be the smallest possible value for h, you know that the proper inequality should be . Clearly the common mistake is to accidentally pick  because you are thinking that the speed must be less than or equal to 70mph. This, however, gives the opposite effect on time so you must flip that sign.

Example Question #2 : Translating Words To Inequalities

Which of the following inequalities properly expresses the following relationship: Tina won more than twice as many medals as George? Let  represent the number of medals Tina won, and  represent the number of medals George won.

Possible Answers:

Correct answer:

Explanation:

To translate this appropriately, it can be helpful to slightly rephrase the topic from "medals won" to "total number of medals is."  If you say it as "Tina's total is more than twice George's total" the words directly line up to math symbols:

Tina's total: 

Is more than: 

George's total: 

So if Tina's total is more than twice George's total, that means that 

Note on this problem that the answer  is true - we know that Tina has more medals than George, so if we double her number of medals it's even a bigger mismatch. But it does not directly translate to the situation provided, so it's not the correct answer here as it does not perfectly answer the exact question.

Example Question #1 : Translating Words To Inequalities

Tim needs to purchase at least 20 breakfast treats for a meeting, and can choose from an assortment of scones , which cost $2.50 and muffins , which cost $3.00. He must purchase more scones than muffins, and he is not allowed to spend more than $56. Which of the following systems of equations accurately represents these conditions?

Possible Answers:

Correct answer:

Explanation:

This problem rewards your knowledge of both inequalities in general and the "or equal to" case specifically. The first mathematical statement you're given is that Tim must purchase "at least 20" items. "At least" means that it could be equal to, but also could be greater than, so you'd use the greater-than-or-equal-to sign, . This translates to .

The second mathematical statement you're given is that Tim must purchase more muffins than scones. Note that it is "more than" and not "at least as many as" so the correct sign to use is . You can set this up as , although this quickly translates to  as well.

The final statement is about cost. Since each scone costs $2.50 and each muffin costs $3, the costs can be summarized as . And you need this to be "no more than" $56.  "No more than" means that it could be equal to $56, or anything less, so you'll use the less-than-or-equal-to sign . This can be expressed as .

Example Question #3 : Translating Words To Inequalities

At a carnival, Clara is allowed to spend up to $20 on games and rides, and each game costs $1 to play and each ride costs $2.50 to ride. If  is the number of games she plays and  is the number of rides she rides, which of the following inequalities properly expresses these conditions?

Possible Answers:

Correct answer:

Explanation:

The amount of money that Clara spends will be a combination of the $1 she spends on each game and the $2.50 she spends on each ride. So her total expenditure can be expressed as , or more simply as .

Her constraint is that she is allowed to spend "up to $20" which means that she can spend any amount less than or equal to $20. So you'll use the less-than-or-equal-to sign, .

Therefore the correct answer is .

Example Question #3 : Translating Words To Inequalities

For a school fundraiser, Leslie sold boxes of apples and pears. Each box of apples cost $12, and each box of pears cost $20. She didn't reach her goal of selling 100 total boxes, but she did break the school record for total sales revenue, exceeding the previous record of $1,600. If  represents the number of boxes of apples she sold and  represents the number of pears she sold, which of the following systems of equations expresses these conditions?

Possible Answers:

Correct answer:

Explanation:

The two relationships you need to express are:

1) She sold fewer total boxes than her goal of 100, so her total  was less than . This is .

2) The total amount sold was greater than $1600. Her total amount sold is the price per box times the number of that box, or . If that was greater than $1600, your inequality is then .

Example Question #4 : Translating Words To Inequalities

Bernard works at two part-time jobs to help pay his college tuition. His goal is to earn at least $300 per week to put toward tuition, but also to work less than 20 hours per week so that he has time to study. He is paid $15 per hour working at the cafeteria, and $18 per hour conducting campus tours. If the number of hours he spends working at the cafeteria each week is  and the number of hours he spends conducting campus tours is , which of the following systems of inequalities expresses these circumstances?

Possible Answers:

Correct answer:

Explanation:

There are two conditions to account for in this problem:

1) Bernard wants to work less than 20 hours. That means his total number of hours at both jobs  must be less than 20, which displays as .

2) Bernard wants his income to be at least $300. "At least" means "greater than or equal to" so you'll use the  sign. And his income will be his rate at each job times the number of hours he works at each. This makes this inequality 

Example Question #5 : Translating Words To Inequalities

Darius is a car salesman whose dealership sells two types of vehicles, cars and trucks. Each car sells for $20,000 and each truck sells for $30,000. For a certain month, Darius had a goal of selling at least 50 total vehicles and selling more than $1,000,000 in sales dollars. If  represents the number of cars he sold that month and  represents the number of trucks he sold that month, which of the following systems of inequalities would express a month in which he did not reach his sales dollars goal but did achieve his total vehicles goal?

Possible Answers:

Correct answer:

Explanation:

This problem has two conditions. One is that Darius did achieve his total sales vehicles goal, and that one was to sell "at least" 50 vehicles. "At least" means greater than or equal to, so we'd use the inequality  here to indicate that he sold 50 or more vehicles to achieve that goal.

The second condition is a goal that he did not hit. The goal was to sell "more than $1,000,000." Here note that if he sold exactly $1,000,000 he would not have reached his goal of "more than $1,000,000" and of course if he sold less than $1,000,000 he would have missed his goal, too. So we can conclude that Darius sold less than or equal to $1,000,000. This means that you'd use the inequality .

Example Question #1 : Translating Words To Inequalities

At his part-time job, Rashid earns $20 per hour. Last week he worked  hours and earned less than $650. Which of the following inequalities properly describes these conditions?

Possible Answers:

Correct answer:

Explanation:

Rashid's pay has two components: his earnings per hour, which we know is $20, and the number of hours he works, . To get his total pay, you multiply those together, so his total earnings are . We know that his total earnings were less than  $650, so we can use the inequality:

Example Question #6 : Translating Words To Inequalities

Milt works at a job that pays him $12 per hour and Caitlyn works at a job that pays her $15 per hour. If Milt works  hours per week and Caitlyn works  hours per week, which of the following inequalities would express that Milt earns at least as much each week as Caitlyn does?

Possible Answers:

Correct answer:

Explanation:

One important consideration for this problem is which inequality sign to use. Since Milt earns "at least as much" as Caitlyn, you should see that "at least" means "greater than or equal to," leading to this sign: 

You then have two total amounts: Milt's $12 per hour times his  hours per week, and Caitlyn's $15 per hour times her  hours per week.  Those are then , which is greater than equal to Caitlyn's total of . The correct answer is then .

Example Question #4 : Translating Words To Inequalities

Karim scored more than twice as many points as Scott.

If k represents the number of points Karim scored and s represents the number of points Scott scored, which of the following inequalities properly represents the relationship above?

Possible Answers:

k > 2s

2k < s

k + 2 > s

k > s + 2

Correct answer:

k > 2s

Explanation:

Word problems often have fairly direct translations to mathematical symbols. If you express the given information as "Karim's total is greater than twice Scott's total" you can directly use symbols to replace words. Karim's total is k, "is greater than" is >, "twice" means "2 multiplied by" and Scott's total is s.  This gives you the correct answer:

k > 2s

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