Question 1
Two competing cell phone plans charge data overage fees of 0.025 dollars per MB and 2.8% of a dollar per MB, respectively. What is the difference between these rates expressed as a percentage of a dollar?
- 2.5%
- 0.53%
- 0.3%
- 0.28%
Explanation: When comparing rates that are expressed in different formats, you need to convert them to the same units before finding the difference. Here you have one rate as a decimal (0.025 dollars per MB) and another as a percentage (2.8% of a dollar per MB).
First, convert both rates to the same format. Since the question asks for the answer as a percentage, let's convert the decimal to a percentage. To convert 0.025 dollars to a percentage of a dollar, multiply by 100: 0.025×100=2.5%
Now you can compare the two rates directly: 2.5% per MB versus 2.8% per MB. The difference is 2.8%−2.5%=0.3%
Looking at the wrong answers: Choice A (2.5%) gives you just the first rate converted to a percentage, but doesn't calculate the difference. Choice B (0.53%) likely comes from incorrectly converting 2.8% to a decimal (0.028) and then subtracting 0.025, giving 0.003, which someone might mistakenly read as 0.53%. Choice D (0.28%) appears to come from confusing 2.8% with 0.28% during the calculation.
The correct answer is C.
Study tip: When comparing rates in different formats, always convert to the same units first. Remember that to convert a decimal to a percentage, you multiply by 100, and to convert a percentage to a decimal, you divide by 100. Double-check your conversions before doing any arithmetic. Question 2
A basketball player's free throw percentage improved from 0.72 to 78%. By how many percentage points did their free throw percentage increase?
- 5.4 percentage points
- 8.3 percentage points
- 0.06 percentage points
- 6 percentage points
Explanation: When you encounter percentage problems, always check whether the given values are in the same format. Here you're comparing a decimal (0.72) to a percentage (78%), so your first step is converting them to the same units.
To find the increase, convert 0.72 to a percentage by multiplying by 100: 0.72×100=72%. Now you can subtract: 78%−72%=6 percentage points.
The correct answer is D) 6 percentage points because this represents the actual difference between the two percentages when expressed in the same units.
Let's examine why the other choices are wrong. Choice A) 5.4 percentage points likely comes from incorrectly calculating 78−72.6 if someone mistakenly converted 0.72 to 72.6%. Choice B) 8.3 percentage points might result from subtracting 78−69.7 or from some other conversion error. Choice C) 0.06 percentage points represents the difference if you left everything in decimal form (0.78−0.72=0.06), but this gives you the answer in decimal form, not percentage points.
Remember that "percentage points" refers to the arithmetic difference between two percentages. This is different from "percent change," which would involve division. When a question asks for an increase in percentage points, simply subtract the percentages after ensuring they're in the same format. Always double-check your unit conversions—decimal 0.72 equals 72%, not 7.2% or 72.6%. Question 3
A store offers a discount where customers save 83 of the original price. If this discount is equivalent to saving 37.5 cents on every dollar spent, what percentage discount does the store offer?
- 37.5%
- 62.5%
- 3.75%
- 38.5%
Explanation: Convert the fraction to a decimal: 3/8 = 0.375. Convert to percentage: 0.375 = 37.5%. The problem confirms this with '37.5 cents on every dollar,' which means 37.5/100 = 37.5%. Choice B incorrectly calculates 100% - 37.5% = 62.5% (the remaining amount, not the discount). Choice C results from moving the decimal point incorrectly (0.375 → 3.75%). Choice D comes from rounding 37.5% incorrectly to 38.5%.
Question 4
Maria calculated that she spent 0.275 of her monthly budget on groceries and 18% of her budget on entertainment. What is the difference between these two expense categories expressed as a percentage?
- 9.5%
- 10.7%
- 7.5%
- 12.3%
Explanation: First, convert 0.275 to a percentage: 0.275 = 27.5%. Then find the difference: 27.5% - 18% = 9.5%. Choice B results from incorrectly adding the percentages (27.5% + 18% = 45.5%, then mistakenly dividing by some factor). Choice C comes from the error 0.275 - 0.18 = 0.095, then converting incorrectly as 0.095 × 100 = 9.5, but reporting 7.5%. Choice D results from calculation errors in the conversion process.
Question 5
A survey found that 0.064 of students prefer math, while 6.8% prefer science. If 250 students were surveyed, how many more students prefer science than math?
- 4 students
- 1 student
- 16 students
- 17 students
Explanation: When you encounter problems mixing decimals and percentages, your first step is always to convert everything to the same format so you can make accurate comparisons.
Let's convert both values to decimals first. Science preference is given as 6.8%, which equals 6.8÷100=0.068. Math preference is already in decimal form: 0.064.
Now calculate how many students prefer each subject out of 250 total students:
- Math: 0.064×250=16 students
- Science: 0.068×250=17 students
The difference is 17−16=1 more student preferring science than math, making B correct.
Let's examine why the other answers are wrong. Answer A (4 students) likely comes from incorrectly calculating the percentage difference (6.8% - 6.4% = 0.4%, then mistakenly treating 0.4% as 4 students). Answer C (16 students) is simply the number who prefer math, not the difference between the two groups. Answer D (17 students) is the number who prefer science, again missing that the question asks for the difference.
The key strategy here is to always convert percentages and decimals to the same format before doing any calculations. Also, pay close attention to what the question is actually asking for—in this case, it's the difference between two groups, not the size of either individual group. Double-check your final step to ensure you're answering the right question. Question 6
A manufacturing process has a defect rate of 2007. If the company wants to report this as a percentage with one decimal place, and the current rate needs to increase by 0.015 to reach their target defect rate, what will the target rate be as a percentage?
- 3.5%
- 5.0%
- 2.0%
- 4.6%
Explanation: When you encounter problems involving fractions, decimals, and percentages together, you need to convert everything to the same format to work with the numbers effectively.
Start by converting the current defect rate 2007 to a decimal: 2007=0.035. The problem states this rate needs to increase by 0.015 to reach the target, so the target rate as a decimal is 0.035+0.015=0.050.
To convert this decimal to a percentage, multiply by 100: 0.050×100=5.0%. This confirms that B) 5.0% is correct.
Now let's examine why the other answers are wrong. Choice A) 3.5% represents the current defect rate, not the target rate—this is what you'd get if you only converted 2007 to a percentage without adding the increase. Choice C) 2.0% might result from calculation errors, perhaps subtracting 0.015 instead of adding it, or from mishandling the decimal conversions. Choice D) 4.6% doesn't correspond to any logical step in this problem and likely represents a computational mistake.
The key strategy here is to work systematically: convert the fraction to decimal form first, then perform the addition, and finally convert to percentage. Many students rush and try to work with mixed formats, leading to errors. Always convert to a common format (usually decimals) before performing arithmetic operations, then convert to the requested format at the end. Question 7
During a science experiment, a solution decreased in concentration from 0.84 to 0.672. What was the percent decrease in concentration?
- 19.6%
- 16.8%
- 25%
- 20%
Explanation: When you encounter percent change problems, you're looking at how much a quantity has increased or decreased relative to its original value. The key formula is: percent change = original valueamount of change×100
Let's work through this step by step. The concentration started at 0.84 and ended at 0.672. First, find the amount of decrease: 0.84−0.672=0.168. Next, divide this change by the original value: 0.840.168=0.20. Finally, convert to a percentage: 0.20×100=20%. This confirms answer D is correct.
Now let's examine why the other choices are wrong. Choice A (19.6%) likely comes from a calculation error, possibly from rounding too early in the process or making an arithmetic mistake. Choice B (16.8%) is a classic trap - this is what you'd get if you mistakenly divided the change (0.168) by the final value (0.672) instead of the original value. Choice C (25%) represents another common error, perhaps from incorrectly calculating the amount of change or using the wrong denominator.
Remember this key strategy: always use the original value as your denominator in percent change problems, not the final value. Also, when dealing with decimals, convert your final decimal answer to a percentage by multiplying by 100. Double-check your arithmetic, especially when subtracting decimals - small errors early in the calculation can lead to significantly wrong answers. Question 8
A stock price changed from 24.50to26.95. If this change represents a 0.1 increase when expressed as a decimal, what was the original stock price that this decimal increase was calculated from?
- $2.45
- $26.95
- $24.50
- $245.00
Explanation: This question tests your understanding of decimal notation and how to interpret problem statements carefully. When you see "decimal increase," you need to identify what baseline value that decimal was calculated from.
Let's work through the logic step by step. The stock price changed from 24.50to26.95, which is an actual dollar increase of 26.95−24.50=2.45. The problem states this change "represents a 0.1 increase when expressed as a decimal." This means that $2.45 equals 0.1 (or 10%) of some original baseline price.
To find that baseline price, we set up the equation: 0.1×baseline=2.45. Solving for the baseline: baseline=0.12.45=24.50. This confirms that the decimal increase was calculated from the original stock price of $24.50.
Looking at the wrong answers: Choice A (2.45)representstheactualdollarchange,notthebaselineforthepercentagecalculation.ChoiceB(26.95) is the final stock price after the increase, not the reference point for calculating the decimal increase. Choice D (245.00)wouldonlymakesenseifyoumisplacedadecimalpoint—ifthebaselinewere245, then 0.1 × 245=24.50, not $2.45.
The correct answer is C ($24.50).
Study tip: When dealing with percentage or decimal increases, always identify whether the question asks for the baseline amount, the change amount, or the final amount. The decimal increase is always calculated from the original value, not the final value. Question 9
A fitness tracker shows that Jake burned calories during different activities as follows: running burned 0.38 of his total calories, weightlifting burned 147 calories out of his 420 total calories, and walking burned the remaining calories.
What percentage of Jake's total calories did he burn while walking?
- 27%
- 35%
- 38%
- 73%
Explanation: First, convert running to percentage: 0.38 = 38%. Then convert weightlifting: 147 ÷ 420 = 0.35 = 35%. Total from running and weightlifting: 38% + 35% = 73%. Walking calories: 100% - 73% = 27%. Choice B gives the weightlifting percentage. Choice C gives the running percentage. Choice D gives the combined running and weightlifting percentage instead of just walking.