Question
CELL PHONE SALES BY COLOR
| Color | Percent of Cell Phones Sold |
|---|---|
| White | |
| Black | 30 |
| Blue | |
| Red | |
| Total | 100 |
The table above shows cell phone sales by color. What percent of the cell phones sold were blue?
- A18%
- B20%
- C22%
- D28%
Show the answer
Correct answer: B
First, use the information given in the table and properties of equations to determine the value of k.
Then determine the percentage of cell phones sold that were blue by finding the value of the expression (given in the table) when .
The percentage of cell phones sold that were blue is 20%.
Question analysis
This algebra question gives cell phone sales by color in a table, with three of the percents written in terms of k: 8k for white, 30 − 2k for blue and k + 5 for red, while black is 30 and the total is 100. Students build an equation, find k, and use it to work out the blue percent, taking care with the negative term.
- Grade level
- Grade 7 content, by NYS standard
- Compared with grade 7
- At grade 7 Grade 7 classwork usually supplies the equation. Here students must notice that the four color entries make up the whole, write and simplify a sum with a negative term, solve it, and evaluate one expression: familiar steps, a little longer than usual.
- Estimated difficulty
- Moderate: several steps, or a close reading with one strong distractor.
- Thinking it demands
- Depth of knowledge 2, skills and concepts · mainly apply
- What makes it hard
- Many steps moderate
- Reading a figure, table or graph minor
- Error-prone arithmetic minor
- Solution
- 4 steps
- Reading level
- Question and choices: about grade 3, 36 words.
- Students, last 30 days
- Not enough answers yet to show (it takes 10).
Why it is hard, traps and wrong answers (shows the answer)
Why it is hard
Once the Total row becomes an equation the algebra is short, so the question mostly tests whether students see that step. Finding k is a halfway point but not a listed choice, which removes the usual early-stop trap. Students who cannot build the equation are left with the blue entry itself, and reading 2k as 2 gives the only wrong answer with a real route; the other two sit two points either side of the key and match no common slip.
Many steps moderate Working it out
Four moves in a row: add the four color entries, set the sum equal to the Total row's 100, solve 7k + 65 = 100 for k, then put k into the blue entry. (Daniel & Embretson, 2010; Enright, Morley & Sheehan, 2002)
Reading a figure, table or graph minor Understanding the question
No equation is printed; it comes from seeing that the table's four percent entries, three of them written in k, must add up to the Total of 100. (Lesh, Post & Behr, 1987; Kirsch & Mosenthal, 1990)
Error-prone arithmetic minor Working it out
Collecting like terms means keeping the blue row's negative term, 8k − 2k + k = 7k, and gathering the constants 30, 30 and 5 into 65. (Brown & Burton, 1978; Junker & Sijtsma, 2001)
Strategy: Since the color percents make up the whole, set the sum of the four entries equal to 100, gather the k terms and the constants, solve for k, and evaluate the blue entry.
Traps
- Numbers taken from the question: Combines the numbers in the question without a model of the situation, or is simply a number that appears in the question.
The wrong answers, classified
A Little reason to choose it (Haladyna & Downing, 1993; Rodriguez, 2005)
Would need k = 6, which makes the four entries total 107 rather than 100; no usual slip in setting up or solving gives that, so it draws guesses.
C Little reason to choose it (Haladyna & Downing, 1993; Rodriguez, 2005)
Would need k = 4, which makes the entries sum to only 93; no common error in combining terms or solving produces it, so it mostly catches guessers.
D Numbers taken from the question (Hegarty, Mayer & Monk, 1995)
Skips solving and reads the blue entry 30 − 2k as 30 − 2, treating k as if it were 1, which gives 28.
Worked solution, step by step
Formulas used
- Combining like terms
- k is the unknown number. a and b are the numbers multiplied by k, called coefficients. Terms with the same letter are like terms. You may add their coefficients, since a groups of k plus b groups of k make a + b groups of k.
- Keeping an equation balanced
- a and b stand for the two sides of an equation. c is any number, but not 0 when you divide. An equation works like a scale in balance. If you do the same thing to both sides, it stays balanced. Use this to get k by itself.
Sort out the table
The table gives the percent of phones sold in each color. Three of the entries use k, a letter that stands for an unknown number. means 8 times k.
You need the percent that were blue, . To find it, you first need k.
Write an equation for the total
Every phone sold was one of the four colors. So the four percents must add up to the whole, 100. The Total row says so.
Combine like terms
Group the k terms together and the plain numbers together. A k on its own means 1k.
Now the equation is . Watch out: the blue row takes away 2k. Keep its minus sign, or the k terms come out wrong.
Solve for k
Subtract 65 from both sides. Then divide both sides by 7.
So k is 5. That is not the answer yet. You still need the blue row.
Find the blue percent
Put 5 in place of k in the blue row.
So 20% of the phones sold were blue. Watch out: 2k means 2 times k, not just 2. Reading as gives 28%, choice D.
So the answer is B, 20%.
Check your answer
Put k = 5 into every row. White: . Black: 30. Blue: 20. Red: . Now add them: . The total matches the table, so k = 5 and 20% are right.
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