MATH 1030-003
Homework #1 Solution
1. Three kinds of apples are all mixed up in a basket. How many apples must you draw (without
looking) from the basket to be sure of getting two apples of one kind?
4. because if you pick less than 4, it’s possible that they are all different.
2. There are 150 people in a class. If 80% of them are registered, how many are not registered?
20% of 150 = (.2)(150) = 30
3. Express “three-fifths” as a fraction, a decimal, and as a percentage.
3
5
= .6 = 60%
4. Evaluate each of the following if a = 4, b = 2/5, and c = −6 :
2
5
2 30
−28
−112
2
)=
= −22.4
a(b + c) = 4( − 6) = 4( − 6 · ) = 4( − ) = 4(
5
5
5
5
5
5
5
a
2
4
8
5
ab + c( ) − c = 4( ) − 6(
) − (−6) = − 6(4 · ) + 6
b
5
2/5
5
2
8
8
5
8 270
262
= − 60 + 6 = − 54 · = −
=−
= −52.4
5
5
5
5
5
5
5b − 3c2 = 5 ·
2
− 3(−6)2 = 2 − 3(36) = 2 − 108 = −106
5
5. Evaluate the following expressions on your calculator:
250
•
27 = 75
34 + 56
• 23 ·
5
+ 6.3 (45 ) = 6, 567.62857
7
√
√
• 3 32 − 15 = 13.09758
1
6. Simplify:
•
x5 x2
x7
=
= x7 x−(−3) = x7 x3 = x10
x−3
x−3
• (x−2 y 3 )2 = (x−2 )2 (y 3 )2 = x−4 y 6
• (x−5 y 4 )2 (x0 y −2 )2 = (x−10 y 8 )(x0 y −4 ) = x−10 y 4
7. If there are 0.82 U.S. dollars in one Canadian dollar, which is smaller–one U.S. dollar, or one
Canadian dollar?
a canadian dollar is smaller because only 82 cents in u.s. dollars fits in a canadian dollar. that is,
one canadian dollar is 82 u.s. cents.
8. One number is 6 times a second number. Find the numbers if their difference is 102.
let x be the first number and let y be the second number. then we know that
x = 6y and x − y = 102.
then plug in x = 6y into x − y = 102 to get
x − y = 102
6y − y = 102
5y = 102
102
y=
= 20.4
5
then plug in y = 20.4 into
x = 6y = 6 · 20.4 = 122.4
so x = 122.4 and y = 20.4.
9. If you drive at an average speed of 65 miles per hour, how long will it take you to drive 530
miles? If you can bike a distance of 45 miles in three hours and 15 minutes, what is your average
biking speed in miles per hour?
we know that distance = rate · time. in the first part, distance = 530 and rate = 65. that gives
the equation
530 = 65t
530
=t
65
so t = 8.154 hours.
2
in the second part, distance = 45 and time = 3.25. so
45 = 3.25r
45
=r
3.25
so r = 13.846 miles per hour.
10. The length of a rectangle is 14 inches more than its width. If the area is 72 square inches, find
the length and width of the rectangle.
let the width of the triangle be x. then the length, which is 13 inches longer than the width is
x + 14. the area of a rectangle is length times width, so the equation is
area = (length) · (width)
72 = (x + 14)(x)
72 = x2 + 14x
0 = x2 + 14x − 72
0 = (x + 18)(x − 4)
which means that x is either −18 or 4. we will use the positive number because we want the width
and length of the rectangle to be positive numbers, so x = 4. this gives that the rectangle has
width of 4 inches and length of 18 inches.
11. Suppose that three-quarters of the freshmen live in a dorm. If two-thirds of the freshmen dorm
residents are women, what percentage of the freshman class are women who live in the dorm?
we want freshmen who live in the dorm and who are women. we know that 34 of freshmen live in
the dorm. and we know that 23 of them are women. so we know that 23 of 34 of freshmen are both
dorm residents and women.
2 3
2
1
· = =
3 4
4
2
so half of the freshmen are women who live in the dorm.
12. Solve for x in the following equations:
3x − 5 = 9 + 7x
3x − 5 + 5 = 9 + 7x + 5
3x = 14 + 7x
3x − 7x = 14 + 7x − 7x
−4x = 14
14
x=−
4
7
x=−
2
3
x2 − 5 = 31
x2 − 5 + 5 = 31 + 5
x2 = 36
√
x = ± 36
x = ±6
x2 − x − 12 = 0
(x − 4)(x − 3) = 0
x = 4 or x = 3
x−3
x
=
5
2
x−3
x
5·2·
= ·5·2
5
2
2(x − 3) = 5x
2x − 6 = 5x
−6 = 3x
6
− =x
3
−2 = x
|x + 3| = 10
x + 3 = 10 or x + 3 = −10
x = 7 or x = −13
4
13. Solve for x and y :
3x − 2y = 5
3x = 2x + 5
2x + 5
x=
3
3x − 2y = 5
−2y = −3x + 5
−3x + 5
y=
−2
x=7−y
y =7−x
14. Graph the line 5x − 2y = 6. What is the y−intercept?
if you have two points, there is only one line that runs through both points. so i think the best
thing to do is find two points on the graph and draw the line between them. i personally think
the two easiest points to find and graph are the x-intercept and the y-intercept. the x-intercept is
found by letting y = 0 and solving for x. if we let y = 0, we have the equation:
5x − 2 · 0 = 6
5x = 6
x = 1.2
the y-intercept is found by letting x = 0 and solving for y.
5 · 0 − 2y = 6
−2y = 6
y = −3
5
15. A warehouse may contain bicycles, tricycles, and cars, and altogether there are 18 wheels in the
warehouse. How many bicycles, tricycles, and cars are there? Give as many answers as possible.
tricycles
6
4
4
2
2
2
2
0
0
0
0
0
bicycles
0
3
1
6
4
2
0
9
7
5
3
1
cars
0
0
1
0
1
2
3
0
1
2
3
4
6
16. A playground is in the shape of a rectangle with a semicircle attached at the shorter end.
Suppose also that the longer side of the rectangle is twice the length of the shorter side and that
the radius of the semicircle is 12 feet. What is the perimeter and the area of the playground?
since the radius of the semicircle is 12 ft, then the diameter is 24 ft and since the semicircle is
attached to the rectangle, then the length of the shorter side is 24 ft. the longer side is twice as
long, so it is 48 ft long.
the perimeter of the whole thing is the combined length of the three sides plus the length of the
arc. the circumference of a circle of radius r is circumference = 2πr, so the length of the arc is
1
· 2π · 12 = 12π.
2
that gives
perimeter = 24 + 48 + 48 + 12π = 120 + 12π.
the area of the whole thing is the combined area of the rectangle and the semicircle. the area of a
rectangle is length times width, so the
area(rect) = 24 · 48 = 1152.
the area of the semicircle is half the area of a circle. the area of a circle of radius r is πr2 . so the
area of the semicircle is
144
1
π = 72π.
area(semicircle) = π · 122 =
2
2
the combined area is
area = 1152 + 72π.
7
17. Assume that the ratio of undergraduate students to graduate students in an institution is 18:7.
What percentage of the student body are graduate students?
let x be the percentage of undergraduate students and let y be the percentage of graduate students.
then x : y = 18 : 7, or xy = 18
7 . if you cross multiply, you get
7x = 18y
also, since all students are either graduates or undergraduates then x+y = 100. if we solve 7x = 18y
for x, we get
18
x = y.
7
18
then we plug x = 7 y into x + y = 100 for x to get
x + y = 100
18
y + y = 100
7
18
7
y + y = 100
7
7
25
y = 100
7
y = 100 ·
7
25
y =4·7
y = 28
since 100 − 28 = 72, then x = 72. so 72% of the students are undergraduates and 28% of the
students are graduate students.
18. Suppose that your annual tuition as a freshman was $1, 856 and each year tuition has increased
5%. If you are now in your senior year, what is your annual tuition this year?
each year, the tuition is the 105% of the tuition from the year before. to write 105% as a decimal,
we write 1.05. also, the word ‘of’ in math usually means multiplication. so, to get the tuition for
the second year, you take the multiplication from the first year and multiply by 1.05. to get the
tuition for the third year, you take the multiplication from the second year and multiply it by 1.05,
T1 =
1856
T2 =
1856 · 1.05 =
1948.8
etc.
T3 = 1948.8 · 1.05 = 2046.24
T4 = 2046.24 · 1.05 = 2148.55
19. The company you work for was doing poorly two years ago, and as a result everyone took a
10% cut in pay for the last year. The company is doing better now, and the CEO just promised
to raise everyone’s salary by 10% for next year. Does this mean that your salary next year will be
the same as it was two years ago? Explain.
it will not be the same. because your salary becomes 90% of your old salary and then your final
salary is 110% of your lowered salary. so your final salary is 110% of 90% of your original salary.
1.10 · 0.90 = 0.99,
so your final salary is 99% of your original salary.
8