QR Test 1999 - Solutions

welles10 · August 7, 2009, 3:07pm

I'll post solutions to the problems here when I have time to work them out again.

If you have questions about the solutions, please quote the entire question/solution so we all know which question you're referring to.

I help this helps some of you out. Remember: NO calculators are allowed for the exam.

Questions can be found here: <a href="http://www.wellesley.edu/FirstYear/Incoming/QR.pdf%5B/url%5D">http://www.wellesley.edu/FirstYear/Incoming/QR.pdf</a>
I won't be writing out the questions themselves. It takes too long and some questions have graphs.

PS. I'm studying for grad school exams right now, so more math practice is actually pretty useful for me ;)

welles10 · August 7, 2009, 3:28pm

<h2>Question 1 - Solution</h2>

This question asks you to look at a bar graph and interpret the data presented.
The white colored bars represent the numbers of prisoners (in thousands).
The grey colored bars represent the number of prisoners per million people (basically a percentage).

Question: One of the statements (a)-(d) is not supported by Figure 1. Which statement is it?

Answer: Statement (b)

<hr>

Solution:

Statement (a): There are more prisoners in Texas (TX) than the combined total of the other nine states shown by the figure. - TRUE

You want to know if there are more prisoners in TX than all the other states combined. This asks you to compare the number of prisoners, so look at the white bars and their corresponding numbers. You see that TX = 140.7

Then add up the numbers of the white bars for all the other states. I estimated this and got about 97 (29.3 + 21.2 + 15.4 + 0.8 + 5.3 + 1.6 + 1.3 + 3.2).

So, 140.7 (TX) > 97 (all other states)
The statement is TRUE.

Statement (b): The percentage of the population that is in prison in Minnesota (MN)
is more than five times the percentage of the population of North Dakota (ND) that is in prison. - FALSE

This statement asks you to look at percentages so look at the grey bars.
The statement says: (%MN) > 5* (%ND)

From the grey bars, we can see that:
%MN= 11.3 per million people
%ND= 11.2 per million people
These “percentages” are almost exactly the same.

The statement is FALSE.

Statement (c): Although there are twice as many prisoners in West Virginia (WV) as
there are in Maine (ME), the rate (per million) in West Virginia is less than twice the rate (per million) in Maine. - TRUE

This statement can be broken down into two parts:
(1) there are twice as many prisoners in WV as there are in ME
and
(2) the rate (per million) in WV is less than twice the rate (per million) in ME

For the statement to be TRUE, part (1) and part (2) both have to be true.
For the statement to be FALSE, either part (1) or part (2) or both has to be false.

Let’s look at part (1) first:
Part (1) asks about the numbers of prisoners, so look at the white bars.
We see that:
WV = 3.2 thousand
ME = 1.6 thousand

We want to know if “there are twice as many prisoners in WV than in ME”
So, WV = 2ME or 3.2 = 21.6.
Part 1 is true.

Now look at part (2):
This part asks for percentages, so look at the grey bars.
%WV = 17.4 per million
%ME = 12.4 per million

We want to know if “the rate (per million) in WV is less than twice the rate (per million) in ME” or if (%WV) < 2(%ME):
We can see that 17.4 < 212.4, so part 2 is true.

Because both part (1) and part (2) are true, this statement is TRUE

Statement (d): The state with the smallest number of prisoners is North Dakota
(ND), and the state with the next smallest number of prisoners is
Vermont (VT). - TRUE

Again, this statement deals with the number of prisoners, so look at the white bars.
We can see that ND has the smallest number with 0.8
The next smallest number is 1.3 which is the number for VT.

This statement is TRUE.

welles10 · August 7, 2009, 3:38pm

Question 2 Solution

This question refers to two pie charts and asks you to interpret what is shown.

Answers:
Part (a): 45 million
Part (b): 17 million

Solution:

Part (a): How large was the rural population of the U.S. in 1900?

We can see that in 1900, the rural population was 60%.
The total population was 75 million.

How large the rural pop is: 60% of 75 mill, or (3/5)*75 mill = 45 mill

Part (b): The rural population of the U.S. actually grew in size between 1900 and 1990, even though it went down in proportion to the overall population. By how many people did the rural population grow?

This asks for the number of people the rural pop grew by, so they want the difference between the number of rural pop in 1990 and in 1900.

We know that the rural pop in 1900 was 45 mill, from part (a).

The rural pop in 1990:
From pi chart, we can see the rural pop is 25%.
The total pop is 248 mill.
So, the rural pop is 25% of 248 mill, or (1/4)*248 mill = 62 mill

Now, the difference between the rural pop in 1990 and that in 1900 is:
rural pop in 1990 - rural pop in 1900
= 62 mill - 45 mill
= 17 mill

welles10 · August 7, 2009, 3:47pm

Question 3 - Solution

This question asks you to look at a graph of a rectangle.

Answers:
Part a: 2 and 5/16 inches
Part b: 111/64 inches squared

Solution:

Part (a): What is the length of the piece of metal in inches (in)?
Look at the graph. This is a basic question to see if you know how to use a measuring stick. You can see that the length of the rectangle is 2 and 5/16 inches.

Part (b): What is the area?
Area = Length * Width
You already know the length from part (a).
Look at the graph to find the width: Width = 3/4 inches.

Area = 2 and 5/16 * 3/4 = 37/16 * 3/4 = 111/64 inches squared

Note: You can simplify fractions if you want to, but as long as it is right, you will not be marked off – so don’t simplify fractions if you don’t have to.

Always put units.

welles10 · August 7, 2009, 4:07pm

Question 4 - Solutions

This question is a basic unit analysis question.

Answers:
Part (a): 3200
Part (b): 6400

Solutions:
Part (a): What is the land’s area in hectares?

Given:
1 hectare = 100m * 100m
Land = 4km * 8km
1km= 1000m

I have two ways to do this part:
Solution 1:

Unit analysis in one step:
Area of land = 4km8km = 32 km squared
Area of hectare = 100m100m = 10000m^2 = 10^4 m^2

32 km^2 * (1000m / 1km)^2 * (1 hectare / 10000 m^2)
= [32] * [(10^3)^2/ (1)^2] * [1/ (10^4)]
=32 * 10^6 / 10^4
=32* 10^2 = 3200

Solution 2:
area of hectare=10^4 m^2

area of land=4km8km
=4000m8000m
=(410^3) * (810^3)
=32*10^6

So,
[3210^6] / [10^4] = 3210^2 = 3200

Part (b): What is land area in hectares?
Given:
1 acre = 1/640 mile squared
1 hectare = 2.5 acres
land = 5miles * 5miles

Land area = 25 miles squared

25 miles squared * [1 acre/ (1/640 mile squared)] * [1 hectare/ 2.5 acre]
= 10*640
= 6400

welles10 · August 7, 2009, 4:26pm

Question 5 - Solution

Answers:
Part (a): 5.42 trillion
Part (b): 4.19 trillion

Solution:
Given: US debt in 1998 = 5.54 trillion

Part (a): Suppose the debt were reduced by $120 billion. What would the new debt be? Note: In the U.S., 1 trillion equals 1,000 billion.

Given: 1 trill = 1000 bill
So, US debt of 5.54 trill = 5540 bill

5540 bill - 120 bill = 5420 bill or 5.42 trill

Part (b): As of July 1998, the size of the U.S. population was 270 million. Suppose every person in America contributed $5,000 towards paying off the national debt. What would the remaining balance be? Note: In the U.S., 1 billion equals 1,000 million.

Total contribution = $5000 per person * 270 mill people = $ 1,350,000 mill
= 1,350 bill = 1.35 trill

Remaining balance = US debt - contribution
= 5.54 trill - 1.35 trill = 4.19 trill

welles10 · August 7, 2009, 4:40pm

Question 6 - Solutions

This question asks you to look at a table of information and find the relationship between the two variables.

Answer: c=30-2s

Solution:

Given:
Staff meeting = 4 hours
Client meeting = 2 hours
Work week = 60 hours
Each week consists of only staff meetings and client meetings.

Find: c (number of client meetings) in terms of s (number of staff meetings)

Because we know the duration of time for each meeting and that each work week is 60 hours, we can write an equation for each work week:

4s + 2c = 60
This equation is in hours: for every 60 hours, the person can either be in “s” staff meetings at 4 hours each or in “c” client meetings at 2 hours each.

Simplify the equation to solve for c and get: c = 30-2s
Verify with the numbers given in the table.

welles10 · August 7, 2009, 4:49pm

Question 7 -Solution

This question may initially look daunting, but with a closer look, it is just a simple “plug and chug” question.

Answers:
No part (a) included in question
Part (b): 500
Part (c): 2250

Solution:

Given:
L=4500
k=8
tau (the funny looking t) = 3

Part (b): Find population when t=0:
Plug in all the numbers to the equation.
You know that anything raised to the power of 0 is 1, so the denominator of the fraction becomes: 1 + 8(1/2)^(0) = 1 + 8(1/2)*1 = 1+4 = 5

So, pop = 4500/5 = 500

Part (c): Find population when t=9:
Here, the denominator is: 1 + 8(1/2)^(9/3) = 1 + 8(1/2)^3
=1 + 8*(1/8) = 1 + 1 = 2

So, pop = 4500/2 = 2250

welles10 · August 7, 2009, 5:00pm

Question 8-Solution

This question asks you to match histograms to statements.

Answers:
Statement (a): Histogram (i)
Statement (b): Histogram (ii)

Solutions:

From the description of the histograms, we know that the scale is:
-12: very course (boulders/rocks)
0: coarse sand
+14: very fine (clay)

Statement (a):
Details: sandy beach with no rocks; coarse to fine sand grains

From the description, one would guess that there would be little to no negative numbers shown in a corresponding graph (“no rocks”).
“Coarse to fine sand grains” suggests something around 0 and a bit more positive.

This corresponds to histogram (i)

Statement (b):
Details: coarsest and finest materials filtered out –> NO coarse or fine materials
uniform mix –> same

No coarse or fine materials suggests there are no large negative or positive numbers.
Uniform mix suggests the numbers come in all about the same frequency.

This corresponds to histogram (ii)

welles10 · August 7, 2009, 5:12pm

Question 9 - Solution

This question asks you to interpret another graph.

Answers:
Part (a): 4
Part (b): 2

Solution:
Part (a): How many earthquakes were less severe than the Mexico City earthquake in 1985, and yet killed more people?

From the graph, we can see that the Mexico City earthquake had a severity rating of a little higher than 8 and killed about 10^4 people. We want to find the number of earthquakes that had smaller severity ratings (less than 8) and higher numbers of deaths (more than 10^4).

To help you with this, you can either draw lines or boundaries on the graph at those numbers or use the lines given to you in the chart.

We can see that there are 4 earthquakes that meet our criteria:
-T’ang-shan , China (1976)
-Kansu Province, China (1932)
-Guatemala City (1976)
-Southern India (1993)

Part (b): How many of the earthquakes killed at least 100 times as many people as the San Salvador earthquake in 1986?

This is very similar to part (a), except with a different set of criteria.

The San Salvador quake had 10^3 deaths. We want to find the number of earthquakes that killed 100 times more people, or 10010^3 = 10^210^3 = 10^5 deaths.

We can see there are 2 quakes that fit this criteria:
-T’ang-shan, China (1976)
-Kansu Province, China (1920)

welles10 · August 7, 2009, 5:19pm

Question 10 - Solution
This is a word problem.

Answer: p1n1 - p2n2 - p3n3

Solution:
The contractor charges p1 dollars per hour for n1 hours, so the contractor will receive a total of p1*n1 dollars; this is the amount that the contractor initially receives.

The contractor must also pay the plumber and the electrician.
The plumber is paid p2 dollars per hour for n2 hours, or a total of p2n2 dollars.
The electrician is paid p3 dollars per hour for n3 hours, or a total of p3n3 dollars.

So, the amount that the contractor is left with is:
Amount received - amount paid out
= p1n1 - (p2n2 + p3n3) = p1n1 - p2n2 - p3n3

welles10 · August 7, 2009, 5:33pm

Question 11 - Solution

This question is yet another graph interpretation question.

Answers:
Part (a): from 50-45 mg
Part (b): 3 mg/hr

Solution:
Part (a): Which of the following drops in drug level takes the most time?

50-45mg:
We can see that the graph starts off at 50mg, stays there for awhile, and then the drug level slowly drops off. Thus, from the graph alone, we can predict that the drop from 50-45mg will take the most time.

Here is the math:
at 50mg: t=0; at 45mg: t=15
time difference = 15 hours

45-40mg:
at 45mg: t=15; at 40mg: t= about 18
time difference = about 3 hours

40-30mg:
at 40mg: t= about 18; at 30mg: t= about 22
time difference = 4 hours

30-20mg:
at 30mg: t= about 22; at 20mg: t= 24
time difference = 2 hours

20-10mg:
at 20mg: t= 24; at 10mg: t=27
time difference = 3 hours

So, the drop from 50-45 mg takes the largest amount of time.

Part (b):On average, about how fast (in mg/hr) does the drug level drop between hour 20 and hour 30?
At hour 20, drug level= 35
At hour 30, level=5
Difference in level = 5-30 = - 30mg
Difference in time = 30-20 = 10 hours

Drop in mg/hr = 30mg/10hr = 3 mg/hr
We are looking at a drop in level, so the negative sign on the level difference can be disregarded.

If you were to interpret this as an increase in level, then there would be an increase of -3mg/hr.

welles10 · August 7, 2009, 5:51pm

Question 12 - Solution
Another word problem.

Answers:
Part (a): 8640
Part (b): 532500

Solution:
Part (a): Over the past few years the population of Nantucket has grown rapidly. In 1990, its population was 6,000, but by 1996 its population had grown to 7,200. Suppose the population of Nantucket goes on to increase by the same proportion between 1996 and 2002 as it did between 1990 and 1996. How large will it be in 2002? Note: The population is assumed to increase by the same proportion not by the same number of persons.

Proportion of growth from 1990-1996:
(1996 pop - 1990 pop)/ (1990 pop) = (7200-6000)/6000 = 0.2
Nantucket grew by 20% between 1990 and 1996.

Find population in 2002 if it grows by the same proportion from 1996 to 2002.
1996 pop * 1.2 = 7200 * 1.2 = 8640

Part (b): In 1996, the population of Norfolk County was 639,000. Suppose the population of Norfolk County had increased by the same proportion between 1990 and 1996 as Nantuckets population did. How large would Norfolk Countys population have been in 1990?

1996 pop = 639000
Grew by 20% from 1990 to 1996. Find pop in 1990.

pop in 1996 = 1.2 * pop in 1990
639000 = 1.2 * pop in 1990
pop in 1990 = 639000 * (5/6) = 532500

welles10 · August 7, 2009, 5:59pm

Question 13 - Solution

Answers:
Part (a): 30 mpg
Part (b): 340 miles

Solution:
Given: E = fuel efficiency = miles per gallon

Part (a): Suppose the car travels for 4 hours at a rate of 60 mph, and that it burns 8 gallons of gas during this time. Find the value of E, the cars fuel efficiency.

Statistics for car:
Time = 4 hours
Rate = 60 mph

Gallon used = 8 gallons

And we remember the equation, Distance = Rate * Time, or D=RT:
Using D=RT, we can find the distance the car travels:
D=RT
D=60pmh* 4 hours = 240 miles

We are given that E = miles/gallon.
So, E = 240 miles/ 8 gallons = 30 mpg

Part (b): How far must the car travel on 8 gallons of gas in order for the value of E to be 42.5?
We use the equation E = miles/gallon and solve for miles.

E = miles/gallon
42.5 = miles / 8
miles = 42.5 * 8 = 340 miles

welles10 · August 7, 2009, 6:11pm

Question 14 - Solution

Personally not a fan of this question, but maybe that’s just because I don’t like the chemistry context of this one.

Answer: T = 25 + 11Q

Solution:

We are given that:
If 1 calorie of heat energy is added to a 1 gram (g) piece of copper, the temperature of the copper will go up by 11°C.

The temperature of a 1 g piece of copper is 25°C.

So, we know that the initial temperature of a 1g piece of copper will always be 25C.
And we know that if 1 cal of heat energy is added to the copper, then the temp increases by 11C –> so, 1 cal of heat energy for every +11C

We are asked to find a formula for the temperature, T, of the copper after Q calories of heat energy have been added.

So, we have:
Original copper temperature = 25 degrees C
Add Q calories of heat energy
–> this means that we are adding Q*11 degrees C to the copper, since 1 cal of heat energy adds 11C

All together, this makes: T = 25 + 11Q

welles10 · August 7, 2009, 6:34pm

Question 15 - Solution
Match the equation to the description.

Answers:
Statement (a): no match
Statement (b): P4
Statement (c): P3
Statement (d): P2

Solution:

Given equations:
P1 = L + rt
P2 = 3L + 2rt
P3 = L/2 + (r + 250)*t
P4 = L + 250 + rt

L, r: positive constants
P = population
t = time (year)

Statement (a): This population begins (in year t = 0) at the same level as P1.

We can see that P1 starts out at a rate of:
P1 = L + 0 = L
We want to find the population that also starts out at a population of L

P2: P2 = 3L + 0 = 3L
P3: P3 = L/2 + 0 = L/2
P4: P4 = L + 250 + 0 = L + 250
no match: we can see that none of these populations match the initial level of P1:, so the answer is no match

Statement (b): This population grows at the same annual rate as P1.

The rate is the part of the equation that is multiplied to the time portion. There’s probably some technical name for it, which I don’t remember right now.
We can see that P1 grows at a rate of: r

P2: 2r
P3: r + 250
P4: r
no match:

Thus, the answer is P4

Statement (c): This population grows by 250 more animals per year than P1

We know P1 grows at a rate of r, so we are looking for a population that grows at a rate of r + 250.

P2: rate is 2r
P3: r + 250
P4: r –> be careful here: the “250” in this equation refers to a population level that is 250 higher than in P1 – this has nothing to do with the rate
no match:

We can see that the answer is P3

Statement (d): This population begins (in year t = 0) with more animals than P1, and it grows at a faster rate

We know that P1’s population begins at L and has a rate of r.
We are looking for a population with a population beginning at something higher than L and with a rate larger than r.

P2: level = 3L (higher than L1); rate = 2r (higher than L1)
P3: level = L/2 (lower than L1); rate = r + 250 (higher than L1)
P4: level = L + 250 (higher than L1); rate = r (same as L1)
no match:

We can see that the answer is P2

welles10 · August 7, 2009, 6:40pm

Question 16 - Solution

Graph a line.
The graph is given in the answers provided in the booklet.

Quick explanation:
Graph the equation 5R - 12n = 60 on the (n, R) axes and label the n- and R- intercepts.

If this is initially confusing, substitute the (n, R) for (x, y) like on “normal” graphs. This way the equation becomes 5y-12x = 60

We can see then that the equation is the same as y = 12/5x + 12, with a slope of 12/5 and a y-intercept of 12. The x-intercept (when y=0) is therefore at -5.

Happy graphing!

welles10 · August 7, 2009, 6:46pm

Question 17 - Solution

Proportions!

Answers:
Part (a): 45mg
Part (b): 3ml

Solutions:
Given: The concentration of iron sulfide is 15 mg for every 0.6 ml of liquid.

Part (a): If a child takes 1.8 ml of this liquid, how much iron sulfide (in mg) is she receiving?
Set up a proportion to find x (the number of mg the child needs):

x mg/1.8ml = 15 mg/0.6 ml

Solve to get:
x= 45 mg

Part (b): Another child needs 75 mg of iron sulfi de per day. How much of the liquid (in ml) should he be given?
Do the same thing as part (a) but with different numbers, and this time you are solving for the amount of ml.

Set up proportion to solve for y (amount of ml):

75 mg/ y ml = 15 mg/ 0.6 ml

Solve to get:
y= 3 ml

welles10 · August 7, 2009, 6:56pm

Question 18 - Solution

Answers:
Part (a): 46
Part (b): 48

Solution:

Part (a): Find the perimeter of Polygon A.

Polygon A has 6 sides. 4 sides are given.
You need to find the length of the missing two sides: the short one and the long one.
The short one = 8-6 = 2
The long one (bottom of the figure) = 3 + 12 = 15

So, total perimeter = 15 + 6 + 12 + 2 + 3 + 8 = 46

Part (b): Find the perimeter of Polygon B.
You are given that the both figures together make a square. You know that the bottom of Polygon A represents the bottom of the square and has length=15. So, all sides of the square must equal 15.

So, you know that:
top side of B = 15
right vertical line = 15 - 6(right vertical line on A) = 9
all the bottom lines on B are the same as the top lines on A: 12, 2, 3
left vertical line = 15 - 8(left vertical line on A) = 7
So, perimeter of B = 15 + 9 + 12 + 2 + 3 + 7 = 48

helen_mes · August 12, 2009, 4:28pm

OMG!!! THANKS!!! This is incredibly helpful!!!

By the way, which exams are you going to take and what grad program are you applying for? I am curious :)

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