A company with 200 employees grew

2024 Handbook

2024 Test B Math · Question 16 of 57

Question

A company with 200 employees grew in size by 25% between 2005 and 2010. The company grew again by 10% between 2010 and 2015. What was the total number of employees in 2015?

A235
B260
C270
D275

Show the answer

Correct answer: D

First, determine the number of employees in 2010:

$200\times 25\%=200\times 0.25=50$

So the number of employees in 2010 is $200+50=250$ .

Next, determine the number of employees in 2015:

$250\times 10\%=250\times 0.10=25$

So the total number of employees in 2015 is $250+25=275$ .

Question analysis

A percent change word problem: a company of 200 employees grows by 25% from 2005 to 2010 and then by another 10% from 2010 to 2015. Students find the number of employees in 2015. Two percent changes in a row are the tricky part, since each percent must be taken of the right amount.

NYS standards: 7.RP.3 multistep ratio and percent problems
6.RP.3c percent as a rate per 100
Grade level: Grade 7 content, by NYS standard
Compared with grade 7: At grade 7 Percent increase is a grade 7 topic, and two increases in a row are a common grade 7 multistep problem. The numbers are friendly, so the demand matches typical grade 7 work.
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: A very close wrong answer moderate
Turning words into math moderate
Many steps minor
Solution: 4 steps
Reading level: Question and choices: about grade 5, 25 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

Each increase alone is easy, and the whole question is about the base of the second one. Adding the percents into a single 35% feels like a shortcut and lands five employees short, a gap estimation cannot reveal. Working period by period, or multiplying 200 by 1.25 and then by 1.1, keeps each percent attached to the right amount.

A very close wrong answer moderate Choosing the answer
270 and 275 employees are five apart, too close to separate by estimate; only taking the second 10% of 250 rather than of 200 decides between them. (Embretson & Wetzel, 1987; Gorin & Embretson, 2006; Ascalon et al., 2007)
Turning words into math moderate Understanding the question
“Grew again by 10%” refers to the company's size in 2010, so the base for the second percent is the new 250 employees, not the original 200. (Hegarty, Mayer & Monk, 1995; Koedinger & Nathan, 2004)
Many steps minor Working it out
Two percent increases make four operations, a percent and an addition for each period, and the second period starts from the first one's result. (Daniel & Embretson, 2010; Enright, Morley & Sheehan, 2002)

Strategy: Work one period at a time: find the 2010 total by increasing 200 by 25%, then increase that new total by 10%.

Traps

Percent of the wrong amount: Finding the whole from a part, or a percent change measured from the wrong starting amount.
Misunderstands the concept: A wrong belief about a concept: that percents can be added, that area grows like length, that averages can be averaged, that a bigger denominator means a bigger fraction.

The wrong answers, classified

A Misunderstands the concept (Smith, diSessa & Roschelle, 1993; De Bock et al., 1998; Ni & Zhou, 2005)
Reveals a misconception Expected pull: weak
Adds the two percents to the employee count as if they were numbers of people, 200 + 25 + 10 = 235.
B Misunderstands the concept (Smith, diSessa & Roschelle, 1993; De Bock et al., 1998; Ni & Zhou, 2005)
Reveals partial knowledge Expected pull: weak
Finds the 250 employees in 2010 correctly, then adds 10 employees instead of 10% of 250, treating the second percent as a head count.
C Percent of the wrong amount (Parker & Leinhardt, 1995)
Reveals a misconception Expected pull: strong
Takes both increases from the original 200, 50 + 20 = 70 more, missing that the second 10% applies to the 250 employees of 2010.

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Percent increase: $\text{new} = \text{old} + \frac{p}{100} \times \text{old}$; Old is the starting amount, and p is the percent it grows by. First find p percent of the old amount. That is the extra. Then add the extra to the old amount to get the new amount.

Sort out what you know
The company has 200 employees at the start, in 2005. From 2005 to 2010 it grows by 25%. From 2010 to 2015 it grows again, by 10%. You need the number of employees in 2015.
Find the number in 2010
$\text{new} = \text{old} + \frac{p}{100} \times \text{old}$
$200 + \frac{25}{100} \times 200 = 200 + 50 = 250$
In 2010 there were 250 employees. Watch out: 25% does not mean 25 people. Adding 25 people and then 10 people gives 235, choice A.
Grow from the new number
The second increase starts in 2010. So take 10% of 250, not 10% of 200.
$250 + \frac{10}{100} \times 250 = 250 + 25 = 275$
In 2015 there were 275 employees.
Avoid the traps and answer
Watch out: taking both percents of the first 200 gives $200 + 50 + 20 = 270$ , choice C. That misses the growth of the 50 new workers. Adding 10 people instead of 10% gives 260, choice B.
So the answer is D, 275.

Check your answer

Growing by 25% is the same as multiplying by 1.25. Growing by 10% is the same as multiplying by 1.1. $200 \times 1.25 \times 1.1 = 250 \times 1.1 = 275$ . The answer should also be a bit more than 270, since the second 10% is taken of a bigger number.

The original DOE question

Question 73 as printed on page 203 of the DOE handbook. — As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2025 Admissions, page 203, question 73. Open page 203 of the handbook (PDF)