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SOLVED QUESTIONS

1. Reduce the following fractions to the lowest terms.

27/27 = 1 and 135/27 = 5

27/135 = 1/5

Convert the following mixed number into decimal form.

4+2/5 = 22/5

= 4.4

Simplify the following:

1- [ XXXXXXXXXX-0.22)]

= 1- [ XXXXXXXXXX)]

= XXXXXXXXXX

= 0.6607

Change the following percent into decimals.

200%

200/100 = 2

What is a salesperson’s commission on net sales of $16,244 if the commission is paid on

a sliding scale of 8 ¼% on the first $12,000, and 11.5% on any additional net sales?

= Net sales are $16,244

8 ¼% on the first $12,000

11 ½% on any additional net sales

8 ¼% *12000 = 0.0825*12000 = $990.00

11 ½%*4244 = 0.115*4244 = $488.06

XXXXXXXXXX = $1,478.06

The salesperson’s commission on net sales of $16,244 is $1478.06

Express the following as a percent:

0.04

0.04 × 100 = 4

The basic pay categories, hourly rates of pay and the number of employees in each category of a company are shown below:

Category

Hourly pay

No. of employees

Supervisor

$40

2

Assistants

$16

10

Helpers

$14

10

TOTALS

$70

22

What is the average rate of pay per employee?

$40 *2 = $80

$16*10 = $160

$14*10 = $140

XXXXXXXXXX = $380

380/30 employees = $17.273

The average rate of pay is $17.27 per employee.

John invested $12,000 in a business on January 1 and an additional $2400 on April 1. He withdraws $1440 in June 1 and invested $2880 on October 1. What was Brent’s average monthly investment balance for the year?

Date XXXXXXXXXXChange XXXXXXXXXXBalance * XXXXXXXXXXMonths invested = XXXXXXXXXXValue

January 1 XXXXXXXXXX12000 * XXXXXXXXXX = $36,000

April 1 XXXXXXXXXX XXXXXXXXXX14400 * XXXXXXXXXX = $28,800

June 1 XXXXXXXXXX XXXXXXXXXX12960 * XXXXXXXXXX = $51,840

October XXXXXXXXXX XXXXXXXXXX15840 * XXXXXXXXXX = $47,520

XXXXXXXXXXTOTALS 12 $164,160

XXXXXXXXXXAverage monthly investment balance = total weighted value/number of months

= $164.160/12 = $13,680.00

Last week Lisa had gross earnings of $ XXXXXXXXXXLisa receives a base salary of $375 and a commission on sales exceeding her quota of $5000. What is her rate of commission if her sales were $6560?

Lisa’s gross earning last week = $441.30

Lisa’s base salary = $ 375.00

Lisa’s sales last week = $6560.00

$441.30-$375.00 = $66.30 commission

$6560.00/$5000.00 = 0.0425

Lisa’s rate of commission is 4.25%

A store located in Vancouver, B.C., sells a computer for $1800 plus GST and PST. If the same model is sold at the same price in a store in Toronto, Ontario, what is the difference in the prices paid by consumers in the two stores?

Computer sold in Vancouver, BC for $1800 plus GST and PST 5% and 7%

Computer sold in Toronto, ON for $1800 plus HST 13%

BC: $1800+12% = XXXXXXXXXXtaxes = $ 2,016 paid in BC

ON: $1800+13% = $ XXXXXXXXXXtaxes = $2034 paid in ON

$2016-$2034 = $18

Walmart sold a stereo set during a sale for $500. Determine the regular selling price of the set if the price of the set had been reduced by one-third of the original regular selling price.

x-1/3x = $500

2/3x = $500

(3)2/3x = $500(3)

2x = 1500

x = 1500/2 =750

x = 750

A wood piece 100-centimeter long is cut into two pieces so that the longer piece is 10 centimetres longer than twice the length of the shorter piece. What is the length of the longer piece?

x+ (2x+10) = 100

3x = 100-10

3x = 90

x = 90/3

x = 30

30+70 = 100

Simplify:

a. (-18x2y)/(3x)

- 18x2y/3x = -6xy

b. (x-4) (x XXXXXXXXXXx-3) (x+2)

x2-x-4x+4-4x2 +4x+24

x2-4x2-5x+4x+4+24

-3x2 – x +28

Simplify:

XXXXXXXXXX

XXXXXXXXXX5)

1-3125 = -3124

Solve each of the following:

a. 2x+0.03x = 255

2.03 x = 255

x = 255/2.03

x = XXXXXXXXXX

b. 8x-6-2x = 14+4x-6

8x – 6 + 6 – 2x – 4x = 14

8x – 2x – 4x = 14

2x = 14

x = 14/2 = 7

Set up a ratio for each of the following and reduce to lowest terms.

a. 14 hours to 2 days

= 14 hours/48 hours =7 hour/24 hours

b. 60 $ per day for 12 employees for 18 days

Labour per day: Employees: Days

= $60 : 12 : 18

= 10: 2 : 3

The cost of a unit is made up of $6.25 material cost, $4.75 labour cost, and $3.25 overhead. What is the ratio that exists between the three elements of cost?

6.25: 4.75: 3.25 = 625: 475: 325

= 25: 19: 13

Solve:

a:6 =20:40

(a/6) = (20/40)

a = (20X6)/40 = 3

Material cost of a computer is five-eighths of total cost, and labour cost is one-third of material cost. If labour cost is $20, what is the total cost of the computer?

Let material cost be x

1/3 = 20/x

(1/3) x = 20

x = 20*3 = 60

Let total cost be y

5/8 = 60/y

5y = 8(60)

y = 96

Total cost is $96

Your utility bill for April is $170. If you pay after the due date, a late payment penalty of $7.72 is added. What is the percent penalty?

Bill = $170

Penalty = 7.72/170 = 0.045

Percent penalty = 45%

A brokerage firm charges a fee of 1 ¼%. If its fee on a stock purchase was $400, what was the amount of the purchase?

Let the amount of the purchase be x$

1 ¼% of x = 400

.0125 x = 400

x = 32,000

The amount of the purchase was $32,000

In Ontario, car was sold for $14,950 including 15% HST. How much was the sales tax on the car?

Let the price of the car be $x

x + %15of x = 14,950

x + 0.15x = 14,950

1.15x = 14,950

x = 14950/1.15

x = $13,000

sales tax = 15% of 13,000 = 1,950

Sales tax was $1,950

How many Canadian dollars can you buy for US$518 if one Canadian dollar is worth US$0.74?

1/0.74 = X/518

X = 518/0.74 = 700

C$ 700

John annual incomes for 2015 and 2016 were 60,000 and 70,000 respectively. Given that consumer price index for the two years was 110.5 and 109.4 respectively. Compute John’s real income for 2015 and 2016.

2015 = 60000/110.5*100 = $54,298.64

2016 = 70000/109.4*100 = $63,985.37

The appraised value of a property has increased since it was purchased by the present owner. The purchase price of the property was $160,000, its appraised value at that time. How much is the current appraised value?

Appraised value:

= $160,000 + $160,000 x

= 160, XXXXXXXXXX,800

Appraised value = $258,800

Media Marketing of Toronto, offers a two-day accommodation coupon for a Hilton resort in Alberta at a promotion price of C$216. If the exchange rate is C$1.18 per U.S dollar, what is the value of the coupon in U.S. dollar?

216 /1.18 = $183.05

The coupon has a value of US$ 183.05

Solve each of the following systems of equations and check your answer.

a. 2X+5y = 0

2X+2y = 6

2X+5y = 0

2X+2y = 6

2X+5y = 0 (1)

2X+2y = 6(2)

Multiply eq (2) by -1 and add to eq (1) to eliminate X

3y = - 6

Y = -6/3 = -2

2X+5y = 0

2X+5(-2) = 0

2X-10 = 0, x = 10/2 = 5

X = 5, Y = - 2

Check:

In Eq (1)

LHS XXXXXXXXXX) = 10 – 10 = 0 = RHS

b. 3x = -2- 3y

5y = 3x-38

3x = - 2 - 3y

5y = 3x - 38

3x = - 2 - 3y(1)

5y = 3x - 38(2)

Re arrange Eq (1) and Eq (2)

3x+3y = -2

-3x+5y = -38

Add both Equations to eliminate x

8y = -40

Y = -40/8 = -5

y = -5

To find x, substitute value of y in Eq (1)

3x = -2 - 3y

3x = -2 – 3(-5)

3x = -2+ 15

x = 13/3

x = 13/3, y = -5

Using algebra, find the slope and y-intercept of the lines represented by the following Equations.

a. 9x+ 3y = 33

9x+ 3y = 33

3y = -9x+33

Y = -3x + 11

Slope, m = -3; y-intercept, b = 11

b. 1-1/2y = 2x

-1/2y = 2x-1

y = - 4x XXXXXXXXXXm = -4; y-intercept = 2

c. (x-2) (y+2) - xy = 8

xy + 2x -2y-4-xy = 8

2x-2y-4 = 8

2x-2y = 8 + 4

-2y = -2x+ 12

y = x – 6

Slope, m = 1; y-intercept, b = -6

A restaurant is offering two dinner specials. The difference between seven times the orders for the first special and four times the orders for the second special is 12. The sum of three-fourths of the orders for the first special and two-thirds of the orders for the second special is 21. Find the number of orders for each special.

First special = x

Second special = y

7x-4y = 12

Multiply by 2

14x-8y = 24 XXXXXXXXXX1)

(3/4) x+(2/3) y = 21

Multiply by 12

9x+8y = 252 XXXXXXXXXX2)

Add both equations

23x = XXXXXXXXXXx = 276/ XXXXXXXXXXx = 12

y = (7x-12)/4 = 18

First = 12

Second = 18

Questions on Logarithm with Solutions

1. Express 53 = 125 in logarithm form.

Solution:

53 = 125

As we know,

ab = c ⇒ logac=b

Therefore;

Log5125 = 3

2. Express log101 = 0 in exponential form.

Solution:

Given, log101 = 0

By the rule, we know;

logac=b ⇒ ab = c

Hence,

100 = 1

3. Find the log of 32 to the base 4.

Solution: log432 = x

4x = 32

(22)x = 2x2x2x2x2

22x = 25

2x=5

x=5/2

Therefore,

log432 =5/2

4. Find x if log5(x-7)=1.

Solution: Given,

log5(x-7)=1

Using logarithm rules, we can write;

51 = x-7

5 = x-7

x=5+7

x=12

5. If logam=n, express an-1 in terms of a and m.

Solution:

logam=n

an=m

an/a=m/a

an-1=m/a

6. Solve for x if log(x-1)+log(x+1)=log21

Solution: log(x-1)+log(x+1)=log21

log(x-1)+log(x+1)=0

log[(x-1)(x+1)]=0

Since, log 1 = 0

(x-1)(x+1) = 1

x2-1=1

x2=2

x=± √2

Since, log of negative number is not defined.

Therefore, x=√2

7. Express log(75/16)-2log(5/9)+log(32/243) in terms of log 2 and log 3.

Solution: log(75/16)-2log(5/9)+log(32/243)

Since, nlogam=logamn

⇒log(75/16)-log(5/9)2+log(32/243)

⇒log(75/16)-log(25/81)+log(32/243)

Since, logam-logan=loga(m/n)

⇒log[(75/16)÷(25/81)]+log(32/243)

⇒log[(75/16)×(81/25)]+log(32/243)

⇒log(243/16)+log(32/243)

Since, logam+logan=logamn

⇒log(32/16)

⇒log2

8. Express 2logx+3logy=log a in logarithm free form.

Solution: 2logx+3logy=log a

logx2+logy3=log a

logx2y3=log a

x2y3=log a

9. Prove that: 2log(15/18)-log(25/162)+log(4/9)=log2

Solution: 2log(15/18)-log(25/162)+log(4/9)=log2

Taking L.H.S.:

⇒2log(15/18)-log(25/162)+log(4/9)

⇒log(15/18)2-log(25/162)+log(4/9)

⇒log(225/324)-log(25/162)+log(4/9)

⇒log[(225/324)(4/9)]-log(25/162)

⇒log[(225/324)(4/9)]/(25/162)

⇒log(72/36)

⇒log2 (R.H.S)

10. Express log XXXXXXXXXXin the form of log10x.

Solution: log10(2+1)

=log102+log101

=log10(2×10)

=log1020

11. Find the value of x, if log10(x-10)=1.

Solution: Given, log10(x-10)=1.

log10(x-10) = log1010

x-10 = 10

x=10+10

x=20

12. Find the value of x, if log(x+5)+log(x-5)=4log2+2log3

Solution: Given,

log(x+5)+log(x-5)=4log2+2log3

log(x+5)(x-5) = 4log2+2log3 [log mn=log m+log n]

log(x2-25) = log24+log32

log(x2-25) = log16+log9

log(x2-25)=log(16×9)

log(x2-25)=log144

x2-25=144

x2=169

x=±√169

x=±13

13. Solve for x, if log(225/log15) = log x

Solution: log x = log(225/log15)

log x=log[(15×15)]/log15

log x = log 152/log 15

log x = 2log 15/log 15

log x = 2

Or

log10x=2

102=x

x=10×10

x=100

Solve:

Solve:

Problems

Problem 1: Find the equation of the line that passes through the points (-1 , 0) and (-4 , 12).

Problem 2: What is the equation of the line through the points (-2 , 0) and (-2 , 4).

Problem 3: Find the equation of the line that passes through the points (7 , 5) and (-9 , 5).

Problem 4: Find the equation of the line through the point (3 , 4) and parallel to the x axis.

Problem 5: What is the equation of the line through the point (-3 , 2) and has x intercept at x = -1.

Problem 6: Find the equation of the line that has an x intercept at x = - 4 and y intercept at y = 5.

Problem 7: What is the equation of the line through the point (-1 , 0) and perpendicular to the line y = 9.

Problem 8: Find the slope, the x and y intercepts of the line given by the equation: -3 x + 5 y = 8.

Problem 9: Find the slope intercept form for the line given by its equation: x / 4 - y / 5 = 3.

Problem 10: Are the lines x = -3 and x = 0 parallel or perpendicular?

Problem 11: For what values of b the point (2 , 2 b) is on the line with equation x - 4 y = 6

Solutions to the Above Problems

Solution to Problem 1:

The slope of the line is given by

m = (y2 - y1) / (x2 - x1) = XXXXXXXXXX) / XXXXXXXXXX)) = - 12 / 3 = - 4

We now write the equation of the line in point slope form: y - y1 = m (x - x1)

y - 0 = - 4(x - (-1))

Simplify and write the equation in general form

y + 4 x = - 4

Solution to Problem 2:

The two points have the same x coordinate and are on the same vertical line whose equation is

x = - 2

Solution to Problem 3:

The two points have the same y coordinate and are on the same horizontal line whose equation is

y = 5

Solution to Problem 4:

A line parallel to the axis has equation of the form y = constant. Since the line we are trying to find passes through (3 , 4), then the equation of the line is given by:

y = 4

Solution to Problem 5:

The x intercept is the point (-1 , 0). The slope of the line is given by:

m = (2 - 0) / XXXXXXXXXX)) = 2 / - 2 = -1

The point slope form of the line is

y - 0 = -1(x - (-1))

The equation can be written as

y = - x - 1

Solution to Problem 6:

The x and y intercepts are the points (-4 , 0) and (0 , 5). The slope of the line is given by:

m = (5 - 0) / XXXXXXXXXX)) = 5 / 4

The point slope form of the line is

y - 5 = (5 / 4)(x - 0)

Multiply all terms by 4 and simplify

4 y - 20 = 5 x

Solution to Problem 7:

The line y = 9 is a horizontal line (parallel to the x axis). The line that is perpendicular to the line y = 9 have the form x = constant. Since the (-1 , 0) is a point on this line, the equation is given by

x = -1

Solution to Problem 8:

To find the slope of the given, we first write in slope intercept form

5y = 3x + 8

y = (3/5) x + 8 / 5

The slope is equal to 3/5. The y intercept is found by setting x = 0 in the equation and solve for y. Hence the y intercept is at y = 8/5. The x intercept is found by setting y = 0 and solve for x. Hence the x intercept is at x = -8/3

Solution to Problem 9:

Given the equation

x / 4 - y / 5 = 3

Keep only the term in y on the left side of the equation

- y / 5 = 3 - x / 4

Multiply all terms by -5

y = (5/4) x - 15

Questions with Solutions

Question 1

Find the slope of a line passing through the points

a. (2 , 3) and (0 , - 1)

b. (- 2 , 4) and (- 2 , 6)

c. (5 , 2) and (- 7 , 2)

Solution to Question 1

a. m = (y2 - y1) / (x2 - x1) = (-1 - 3) / (0 - 2) = 2

b. m = (6 - 4) / XXXXXXXXXX)

The division by -2 + 2 = 0 is undefined and the slope in this case is undefined. The line passing through the given points is a vertical line.

c. m = (2 - 2) / XXXXXXXXXX) = 0

The slope is equal to 0 and the line through the given points is a horizontal line.

Question 2

Find the equation of the line that passes through the point (-2 , 5) and has a slope of -4.

Solution to Question 2

· Substitute y1 , x1 and m in the point slope form of a line

y - y1 = m(x - x1)

y - 5 = - 4(x - (-2))

y = - 4 x - 3

Question 3

Find the equation of the line that passes through the points (0 , -1) and (3 , 5).

Solution to Question 3

· We first calculate the slope of the line

m = XXXXXXXXXX)) / (3 - 0) = 6 / 3 = 2

· Use the slope and any of the two points to write the equation of the line using the point slope form.

y - y1 = m(x - x1)

using the first point

y - (-1) = 2(x - 0)

y = 2 x - 1

Question 4

Find the slope of the line given by the equation

- 2 x + 4 y = 6

Solution to Question 4

· Given the equation

- 2 x + 4 y = 6

· Write the equation in slope intercept form

4 y = 2 x + 6

y = (1 / 2) x + 3 / 2

· The slope of the line is given by the coefficient of x and is equal to 1 / 2.

Question 5

Find an equation of the line that passes through the point (-2 , 3) and is parallel to the line 4 x + 4 y = 8

Solution to Question 5

· Let m1 be the slope of the line whose equation is to be found and m2 the slope of the given line. Rewrite the given equation in slope intercept form and find its slope.

4 y = - 4 x + 8

· Divide both sides by 4

y = - x + 2

slope m2 = - 1.

· Two lines are parallel if and only if they have equal slopes

m1 = m2 = - 1

· We now use the point slope form to find the equation of the line with slope m1.

y - 3 = - 1(x - (-2))

which may be written as

y = - x + 1

Question 6

Find an equation of the line that passes through the point (0 , - 3) and is perpendicular to the line - x + y = 2.

Solution to Question 6

· Let m1 be the slope of the line whose equation is to be found and m2 the slope of the given line. Rewrite the given equation in slope intercept form and find its slope.

y = x + 2

slope m2 = 1

· Two lines are perpendicular if and only their slopes are such that

m1 × m2 = - 1

· This gives m1 = -1

· We now use the point slope form to find the equation of the line with slope m1.

y - (-3) = -1(x - 0)

which may be written

y = - x - 3

Matched Questions

The following are questions matched to the questions presented above. Their answers are also included.

Matched Question 1

Find the slope of a line passing through the points

1. (- 2 , 7) and (- 2 , - 1)

2. (2 , 4) and (- 2 , 6)

3. (- 1 , - 2) and (4 , - 2)

Matched Question 2

line that passes through the point (3 , 0) and has a slope of - 1.

Matched Question 3

Find the equation of the line that passes through the points (2 , 0) and (3 , 3).

Matched Question 4

Find the slope of the line given by the equation

x - 3 y = - 9

Matched Question 5

Find an equation of the line that passes through the point (-1 , 0) and is parallel to the line - 2 x + 2 y = 8.

Matched Question 6

Find an equation of the line that passes through the point (-2 , 1) and is perpendicular to the line x + 2 y = -2.

Answers to the Matched Questions

Matched Question 1

1) undefined

2) - 1 / 2

3) 0

Matched Question 2

y = - x + 3

Matched Question 3

y = 3 x - 6

Matched Question 4

slope = 1 / 3

Matched Question 5

y = x + 1

Matched Question 6

y = 2 x + 5

Question: Find the mean, median, mode, and range for the following list of values:

13, 18, 13, 14, 13, 16, 14, 21, 13

Solution:

Given data: 13, 18, 13, 14, 13, 16, 14, 21, 13

The mean is the usual average.

Mean = XXXXXXXXXX+21+139=15

(Note that the mean is not a value from the original list. This is a common result. You should not assume that your mean will be one of your original numbers.)

The median is the middle value, so to rewrite the list in ascending order as given below:

13, 13, 13, 13, 14, 14, 16, 18, 21

There are nine numbers in the list, so the middle one will be

9+12=102=5

= 5th number

Hence, the median is 14.

The mode is the number that is repeated more often than any other, so 13 is the mode.

The largest value in the list is 21, and the smallest is 13, so the range is 21 – 13 = 8.

Mean = 15

Median = 14

Mode = 13

Range = 8

Practice questions

Q1) The following table represents the distribution of the annual number of days over 100 degrees Fahrenheit for Dallas-Fort Worth for a sample of 80 years from 1905 to 2004. Find sample standard deviation?

Days Above 100 Degrees

Number of Years

0 - 9

25

10 - 19

33

20 - 29

14

30 - 39

5

40 - 49

2

50 - 59

1

Q2) What is the mean of the following numbers? 10, 39, 71, 39, 76, 38, 25.

Q3) What is the median of the following numbers? 10, 39, 71, 42, 39, 76, 38, 25

Q4) The number of service upgrades sold by each of 30 employees is as follows: 32, 6, 21, 10, 8, 11, 12, 36, 17, 16, 15, 18, 40, 24, 21, 23, 24, 24, 29, 16, 32, 31, 10, 30, 35, 32, 18, 39, 12, 20 What is the median number of service upgrades sold by the 30 employees?

Q5) What is the mode of the following numbers? 12, 11, 14, 10, 8, 13, 11, 9

Q6) The following data represents the age distribution of a sample of 100 people covered by health insurance (private or government). The sample was taken in 2003. Find population standard deviation.

Age

Number

25 - 34

23

35 - 44

29

45 - 54

28

55 - 64

20

Q7)The following data represent the high temperature distribution in degrees Fahrenheit for a sample of 40 days from the month of August in Chicago since 1872. Find sample standard deviation.

Temperature

Days

60 - 69

3

70 - 79

15

80 - 89

17

90 - 99

5

Q8)A sample of college students was asked how much they spent monthly on a cell phone plan (to the nearest dollar). Find sample standard deviation.

Monthly Cell Phone Plan Cost ($)

Number of Students

10 - 19

8

20 - 29

16

30 - 39

21

40 - 49

11

50 - 59

4

Q9) The following data represent the difference in scores between the winning and losing teams in a sample of 15 college football bowl games from XXXXXXXXXXFind sample standard deviation.

Point Difference

Number of Bowl Games

1 - 5

8

6 - 10

0

11 - 15

2

16 - 20

3

21 - 25

1

26 - 30

0

31 - 35

1

Q10) Draw stem and leaf plot of:

23,24,25,56,34,66,47,48,49,28, 29, 69,33

Example 1 – Standard deviation

A hen lays eight eggs. Each egg was weighed and recorded as follows:

60 g, 56 g, 61 g, 68 g, 51 g, 53 g, 69 g, 54 g.

a. First, calculate the mean:

b. Now, find the standard deviation.

Table 1. Weight of eggs, in grams

Weight (x)

(x - )

(x - )2

60

1

1

56

-3

9

61

2

4

68

9

81

51

-8

64

53

-6

36

69

10

100

54

-5

25

472

320

c.

Using the information from the above table, we can see that

In order to calculate the standard deviation, we must use the following formula:

1. Reduce the following fractions to the lowest terms.

27/27 = 1 and 135/27 = 5

27/135 = 1/5

Convert the following mixed number into decimal form.

4+2/5 = 22/5

= 4.4

Simplify the following:

1- [ XXXXXXXXXX-0.22)]

= 1- [ XXXXXXXXXX)]

= XXXXXXXXXX

= 0.6607

Change the following percent into decimals.

200%

200/100 = 2

What is a salesperson’s commission on net sales of $16,244 if the commission is paid on

a sliding scale of 8 ¼% on the first $12,000, and 11.5% on any additional net sales?

= Net sales are $16,244

8 ¼% on the first $12,000

11 ½% on any additional net sales

8 ¼% *12000 = 0.0825*12000 = $990.00

11 ½%*4244 = 0.115*4244 = $488.06

XXXXXXXXXX = $1,478.06

The salesperson’s commission on net sales of $16,244 is $1478.06

Express the following as a percent:

0.04

0.04 × 100 = 4

The basic pay categories, hourly rates of pay and the number of employees in each category of a company are shown below:

Category

Hourly pay

No. of employees

Supervisor

$40

2

Assistants

$16

10

Helpers

$14

10

TOTALS

$70

22

What is the average rate of pay per employee?

$40 *2 = $80

$16*10 = $160

$14*10 = $140

XXXXXXXXXX = $380

380/30 employees = $17.273

The average rate of pay is $17.27 per employee.

John invested $12,000 in a business on January 1 and an additional $2400 on April 1. He withdraws $1440 in June 1 and invested $2880 on October 1. What was Brent’s average monthly investment balance for the year?

Date XXXXXXXXXXChange XXXXXXXXXXBalance * XXXXXXXXXXMonths invested = XXXXXXXXXXValue

January 1 XXXXXXXXXX12000 * XXXXXXXXXX = $36,000

April 1 XXXXXXXXXX XXXXXXXXXX14400 * XXXXXXXXXX = $28,800

June 1 XXXXXXXXXX XXXXXXXXXX12960 * XXXXXXXXXX = $51,840

October XXXXXXXXXX XXXXXXXXXX15840 * XXXXXXXXXX = $47,520

XXXXXXXXXXTOTALS 12 $164,160

XXXXXXXXXXAverage monthly investment balance = total weighted value/number of months

= $164.160/12 = $13,680.00

Last week Lisa had gross earnings of $ XXXXXXXXXXLisa receives a base salary of $375 and a commission on sales exceeding her quota of $5000. What is her rate of commission if her sales were $6560?

Lisa’s gross earning last week = $441.30

Lisa’s base salary = $ 375.00

Lisa’s sales last week = $6560.00

$441.30-$375.00 = $66.30 commission

$6560.00/$5000.00 = 0.0425

Lisa’s rate of commission is 4.25%

A store located in Vancouver, B.C., sells a computer for $1800 plus GST and PST. If the same model is sold at the same price in a store in Toronto, Ontario, what is the difference in the prices paid by consumers in the two stores?

Computer sold in Vancouver, BC for $1800 plus GST and PST 5% and 7%

Computer sold in Toronto, ON for $1800 plus HST 13%

BC: $1800+12% = XXXXXXXXXXtaxes = $ 2,016 paid in BC

ON: $1800+13% = $ XXXXXXXXXXtaxes = $2034 paid in ON

$2016-$2034 = $18

Walmart sold a stereo set during a sale for $500. Determine the regular selling price of the set if the price of the set had been reduced by one-third of the original regular selling price.

x-1/3x = $500

2/3x = $500

(3)2/3x = $500(3)

2x = 1500

x = 1500/2 =750

x = 750

A wood piece 100-centimeter long is cut into two pieces so that the longer piece is 10 centimetres longer than twice the length of the shorter piece. What is the length of the longer piece?

x+ (2x+10) = 100

3x = 100-10

3x = 90

x = 90/3

x = 30

30+70 = 100

Simplify:

a. (-18x2y)/(3x)

- 18x2y/3x = -6xy

b. (x-4) (x XXXXXXXXXXx-3) (x+2)

x2-x-4x+4-4x2 +4x+24

x2-4x2-5x+4x+4+24

-3x2 – x +28

Simplify:

XXXXXXXXXX

XXXXXXXXXX5)

1-3125 = -3124

Solve each of the following:

a. 2x+0.03x = 255

2.03 x = 255

x = 255/2.03

x = XXXXXXXXXX

b. 8x-6-2x = 14+4x-6

8x – 6 + 6 – 2x – 4x = 14

8x – 2x – 4x = 14

2x = 14

x = 14/2 = 7

Set up a ratio for each of the following and reduce to lowest terms.

a. 14 hours to 2 days

= 14 hours/48 hours =7 hour/24 hours

b. 60 $ per day for 12 employees for 18 days

Labour per day: Employees: Days

= $60 : 12 : 18

= 10: 2 : 3

The cost of a unit is made up of $6.25 material cost, $4.75 labour cost, and $3.25 overhead. What is the ratio that exists between the three elements of cost?

6.25: 4.75: 3.25 = 625: 475: 325

= 25: 19: 13

Solve:

a:6 =20:40

(a/6) = (20/40)

a = (20X6)/40 = 3

Material cost of a computer is five-eighths of total cost, and labour cost is one-third of material cost. If labour cost is $20, what is the total cost of the computer?

Let material cost be x

1/3 = 20/x

(1/3) x = 20

x = 20*3 = 60

Let total cost be y

5/8 = 60/y

5y = 8(60)

y = 96

Total cost is $96

Your utility bill for April is $170. If you pay after the due date, a late payment penalty of $7.72 is added. What is the percent penalty?

Bill = $170

Penalty = 7.72/170 = 0.045

Percent penalty = 45%

A brokerage firm charges a fee of 1 ¼%. If its fee on a stock purchase was $400, what was the amount of the purchase?

Let the amount of the purchase be x$

1 ¼% of x = 400

.0125 x = 400

x = 32,000

The amount of the purchase was $32,000

In Ontario, car was sold for $14,950 including 15% HST. How much was the sales tax on the car?

Let the price of the car be $x

x + %15of x = 14,950

x + 0.15x = 14,950

1.15x = 14,950

x = 14950/1.15

x = $13,000

sales tax = 15% of 13,000 = 1,950

Sales tax was $1,950

How many Canadian dollars can you buy for US$518 if one Canadian dollar is worth US$0.74?

1/0.74 = X/518

X = 518/0.74 = 700

C$ 700

John annual incomes for 2015 and 2016 were 60,000 and 70,000 respectively. Given that consumer price index for the two years was 110.5 and 109.4 respectively. Compute John’s real income for 2015 and 2016.

2015 = 60000/110.5*100 = $54,298.64

2016 = 70000/109.4*100 = $63,985.37

The appraised value of a property has increased since it was purchased by the present owner. The purchase price of the property was $160,000, its appraised value at that time. How much is the current appraised value?

Appraised value:

= $160,000 + $160,000 x

= 160, XXXXXXXXXX,800

Appraised value = $258,800

Media Marketing of Toronto, offers a two-day accommodation coupon for a Hilton resort in Alberta at a promotion price of C$216. If the exchange rate is C$1.18 per U.S dollar, what is the value of the coupon in U.S. dollar?

216 /1.18 = $183.05

The coupon has a value of US$ 183.05

Solve each of the following systems of equations and check your answer.

a. 2X+5y = 0

2X+2y = 6

2X+5y = 0

2X+2y = 6

2X+5y = 0 (1)

2X+2y = 6(2)

Multiply eq (2) by -1 and add to eq (1) to eliminate X

3y = - 6

Y = -6/3 = -2

2X+5y = 0

2X+5(-2) = 0

2X-10 = 0, x = 10/2 = 5

X = 5, Y = - 2

Check:

In Eq (1)

LHS XXXXXXXXXX) = 10 – 10 = 0 = RHS

b. 3x = -2- 3y

5y = 3x-38

3x = - 2 - 3y

5y = 3x - 38

3x = - 2 - 3y(1)

5y = 3x - 38(2)

Re arrange Eq (1) and Eq (2)

3x+3y = -2

-3x+5y = -38

Add both Equations to eliminate x

8y = -40

Y = -40/8 = -5

y = -5

To find x, substitute value of y in Eq (1)

3x = -2 - 3y

3x = -2 – 3(-5)

3x = -2+ 15

x = 13/3

x = 13/3, y = -5

Using algebra, find the slope and y-intercept of the lines represented by the following Equations.

a. 9x+ 3y = 33

9x+ 3y = 33

3y = -9x+33

Y = -3x + 11

Slope, m = -3; y-intercept, b = 11

b. 1-1/2y = 2x

-1/2y = 2x-1

y = - 4x XXXXXXXXXXm = -4; y-intercept = 2

c. (x-2) (y+2) - xy = 8

xy + 2x -2y-4-xy = 8

2x-2y-4 = 8

2x-2y = 8 + 4

-2y = -2x+ 12

y = x – 6

Slope, m = 1; y-intercept, b = -6

A restaurant is offering two dinner specials. The difference between seven times the orders for the first special and four times the orders for the second special is 12. The sum of three-fourths of the orders for the first special and two-thirds of the orders for the second special is 21. Find the number of orders for each special.

First special = x

Second special = y

7x-4y = 12

Multiply by 2

14x-8y = 24 XXXXXXXXXX1)

(3/4) x+(2/3) y = 21

Multiply by 12

9x+8y = 252 XXXXXXXXXX2)

Add both equations

23x = XXXXXXXXXXx = 276/ XXXXXXXXXXx = 12

y = (7x-12)/4 = 18

First = 12

Second = 18

Questions on Logarithm with Solutions

1. Express 53 = 125 in logarithm form.

Solution:

53 = 125

As we know,

ab = c ⇒ logac=b

Therefore;

Log5125 = 3

2. Express log101 = 0 in exponential form.

Solution:

Given, log101 = 0

By the rule, we know;

logac=b ⇒ ab = c

Hence,

100 = 1

3. Find the log of 32 to the base 4.

Solution: log432 = x

4x = 32

(22)x = 2x2x2x2x2

22x = 25

2x=5

x=5/2

Therefore,

log432 =5/2

4. Find x if log5(x-7)=1.

Solution: Given,

log5(x-7)=1

Using logarithm rules, we can write;

51 = x-7

5 = x-7

x=5+7

x=12

5. If logam=n, express an-1 in terms of a and m.

Solution:

logam=n

an=m

an/a=m/a

an-1=m/a

6. Solve for x if log(x-1)+log(x+1)=log21

Solution: log(x-1)+log(x+1)=log21

log(x-1)+log(x+1)=0

log[(x-1)(x+1)]=0

Since, log 1 = 0

(x-1)(x+1) = 1

x2-1=1

x2=2

x=± √2

Since, log of negative number is not defined.

Therefore, x=√2

7. Express log(75/16)-2log(5/9)+log(32/243) in terms of log 2 and log 3.

Solution: log(75/16)-2log(5/9)+log(32/243)

Since, nlogam=logamn

⇒log(75/16)-log(5/9)2+log(32/243)

⇒log(75/16)-log(25/81)+log(32/243)

Since, logam-logan=loga(m/n)

⇒log[(75/16)÷(25/81)]+log(32/243)

⇒log[(75/16)×(81/25)]+log(32/243)

⇒log(243/16)+log(32/243)

Since, logam+logan=logamn

⇒log(32/16)

⇒log2

8. Express 2logx+3logy=log a in logarithm free form.

Solution: 2logx+3logy=log a

logx2+logy3=log a

logx2y3=log a

x2y3=log a

9. Prove that: 2log(15/18)-log(25/162)+log(4/9)=log2

Solution: 2log(15/18)-log(25/162)+log(4/9)=log2

Taking L.H.S.:

⇒2log(15/18)-log(25/162)+log(4/9)

⇒log(15/18)2-log(25/162)+log(4/9)

⇒log(225/324)-log(25/162)+log(4/9)

⇒log[(225/324)(4/9)]-log(25/162)

⇒log[(225/324)(4/9)]/(25/162)

⇒log(72/36)

⇒log2 (R.H.S)

10. Express log XXXXXXXXXXin the form of log10x.

Solution: log10(2+1)

=log102+log101

=log10(2×10)

=log1020

11. Find the value of x, if log10(x-10)=1.

Solution: Given, log10(x-10)=1.

log10(x-10) = log1010

x-10 = 10

x=10+10

x=20

12. Find the value of x, if log(x+5)+log(x-5)=4log2+2log3

Solution: Given,

log(x+5)+log(x-5)=4log2+2log3

log(x+5)(x-5) = 4log2+2log3 [log mn=log m+log n]

log(x2-25) = log24+log32

log(x2-25) = log16+log9

log(x2-25)=log(16×9)

log(x2-25)=log144

x2-25=144

x2=169

x=±√169

x=±13

13. Solve for x, if log(225/log15) = log x

Solution: log x = log(225/log15)

log x=log[(15×15)]/log15

log x = log 152/log 15

log x = 2log 15/log 15

log x = 2

Or

log10x=2

102=x

x=10×10

x=100

Solve:

Solve:

Problems

Problem 1: Find the equation of the line that passes through the points (-1 , 0) and (-4 , 12).

Problem 2: What is the equation of the line through the points (-2 , 0) and (-2 , 4).

Problem 3: Find the equation of the line that passes through the points (7 , 5) and (-9 , 5).

Problem 4: Find the equation of the line through the point (3 , 4) and parallel to the x axis.

Problem 5: What is the equation of the line through the point (-3 , 2) and has x intercept at x = -1.

Problem 6: Find the equation of the line that has an x intercept at x = - 4 and y intercept at y = 5.

Problem 7: What is the equation of the line through the point (-1 , 0) and perpendicular to the line y = 9.

Problem 8: Find the slope, the x and y intercepts of the line given by the equation: -3 x + 5 y = 8.

Problem 9: Find the slope intercept form for the line given by its equation: x / 4 - y / 5 = 3.

Problem 10: Are the lines x = -3 and x = 0 parallel or perpendicular?

Problem 11: For what values of b the point (2 , 2 b) is on the line with equation x - 4 y = 6

Solutions to the Above Problems

Solution to Problem 1:

The slope of the line is given by

m = (y2 - y1) / (x2 - x1) = XXXXXXXXXX) / XXXXXXXXXX)) = - 12 / 3 = - 4

We now write the equation of the line in point slope form: y - y1 = m (x - x1)

y - 0 = - 4(x - (-1))

Simplify and write the equation in general form

y + 4 x = - 4

Solution to Problem 2:

The two points have the same x coordinate and are on the same vertical line whose equation is

x = - 2

Solution to Problem 3:

The two points have the same y coordinate and are on the same horizontal line whose equation is

y = 5

Solution to Problem 4:

A line parallel to the axis has equation of the form y = constant. Since the line we are trying to find passes through (3 , 4), then the equation of the line is given by:

y = 4

Solution to Problem 5:

The x intercept is the point (-1 , 0). The slope of the line is given by:

m = (2 - 0) / XXXXXXXXXX)) = 2 / - 2 = -1

The point slope form of the line is

y - 0 = -1(x - (-1))

The equation can be written as

y = - x - 1

Solution to Problem 6:

The x and y intercepts are the points (-4 , 0) and (0 , 5). The slope of the line is given by:

m = (5 - 0) / XXXXXXXXXX)) = 5 / 4

The point slope form of the line is

y - 5 = (5 / 4)(x - 0)

Multiply all terms by 4 and simplify

4 y - 20 = 5 x

Solution to Problem 7:

The line y = 9 is a horizontal line (parallel to the x axis). The line that is perpendicular to the line y = 9 have the form x = constant. Since the (-1 , 0) is a point on this line, the equation is given by

x = -1

Solution to Problem 8:

To find the slope of the given, we first write in slope intercept form

5y = 3x + 8

y = (3/5) x + 8 / 5

The slope is equal to 3/5. The y intercept is found by setting x = 0 in the equation and solve for y. Hence the y intercept is at y = 8/5. The x intercept is found by setting y = 0 and solve for x. Hence the x intercept is at x = -8/3

Solution to Problem 9:

Given the equation

x / 4 - y / 5 = 3

Keep only the term in y on the left side of the equation

- y / 5 = 3 - x / 4

Multiply all terms by -5

y = (5/4) x - 15

Questions with Solutions

Question 1

Find the slope of a line passing through the points

a. (2 , 3) and (0 , - 1)

b. (- 2 , 4) and (- 2 , 6)

c. (5 , 2) and (- 7 , 2)

Solution to Question 1

a. m = (y2 - y1) / (x2 - x1) = (-1 - 3) / (0 - 2) = 2

b. m = (6 - 4) / XXXXXXXXXX)

The division by -2 + 2 = 0 is undefined and the slope in this case is undefined. The line passing through the given points is a vertical line.

c. m = (2 - 2) / XXXXXXXXXX) = 0

The slope is equal to 0 and the line through the given points is a horizontal line.

Question 2

Find the equation of the line that passes through the point (-2 , 5) and has a slope of -4.

Solution to Question 2

· Substitute y1 , x1 and m in the point slope form of a line

y - y1 = m(x - x1)

y - 5 = - 4(x - (-2))

y = - 4 x - 3

Question 3

Find the equation of the line that passes through the points (0 , -1) and (3 , 5).

Solution to Question 3

· We first calculate the slope of the line

m = XXXXXXXXXX)) / (3 - 0) = 6 / 3 = 2

· Use the slope and any of the two points to write the equation of the line using the point slope form.

y - y1 = m(x - x1)

using the first point

y - (-1) = 2(x - 0)

y = 2 x - 1

Question 4

Find the slope of the line given by the equation

- 2 x + 4 y = 6

Solution to Question 4

· Given the equation

- 2 x + 4 y = 6

· Write the equation in slope intercept form

4 y = 2 x + 6

y = (1 / 2) x + 3 / 2

· The slope of the line is given by the coefficient of x and is equal to 1 / 2.

Question 5

Find an equation of the line that passes through the point (-2 , 3) and is parallel to the line 4 x + 4 y = 8

Solution to Question 5

· Let m1 be the slope of the line whose equation is to be found and m2 the slope of the given line. Rewrite the given equation in slope intercept form and find its slope.

4 y = - 4 x + 8

· Divide both sides by 4

y = - x + 2

slope m2 = - 1.

· Two lines are parallel if and only if they have equal slopes

m1 = m2 = - 1

· We now use the point slope form to find the equation of the line with slope m1.

y - 3 = - 1(x - (-2))

which may be written as

y = - x + 1

Question 6

Find an equation of the line that passes through the point (0 , - 3) and is perpendicular to the line - x + y = 2.

Solution to Question 6

· Let m1 be the slope of the line whose equation is to be found and m2 the slope of the given line. Rewrite the given equation in slope intercept form and find its slope.

y = x + 2

slope m2 = 1

· Two lines are perpendicular if and only their slopes are such that

m1 × m2 = - 1

· This gives m1 = -1

· We now use the point slope form to find the equation of the line with slope m1.

y - (-3) = -1(x - 0)

which may be written

y = - x - 3

Matched Questions

The following are questions matched to the questions presented above. Their answers are also included.

Matched Question 1

Find the slope of a line passing through the points

1. (- 2 , 7) and (- 2 , - 1)

2. (2 , 4) and (- 2 , 6)

3. (- 1 , - 2) and (4 , - 2)

Matched Question 2

line that passes through the point (3 , 0) and has a slope of - 1.

Matched Question 3

Find the equation of the line that passes through the points (2 , 0) and (3 , 3).

Matched Question 4

Find the slope of the line given by the equation

x - 3 y = - 9

Matched Question 5

Find an equation of the line that passes through the point (-1 , 0) and is parallel to the line - 2 x + 2 y = 8.

Matched Question 6

Find an equation of the line that passes through the point (-2 , 1) and is perpendicular to the line x + 2 y = -2.

Answers to the Matched Questions

Matched Question 1

1) undefined

2) - 1 / 2

3) 0

Matched Question 2

y = - x + 3

Matched Question 3

y = 3 x - 6

Matched Question 4

slope = 1 / 3

Matched Question 5

y = x + 1

Matched Question 6

y = 2 x + 5

Question: Find the mean, median, mode, and range for the following list of values:

13, 18, 13, 14, 13, 16, 14, 21, 13

Solution:

Given data: 13, 18, 13, 14, 13, 16, 14, 21, 13

The mean is the usual average.

Mean = XXXXXXXXXX+21+139=15

(Note that the mean is not a value from the original list. This is a common result. You should not assume that your mean will be one of your original numbers.)

The median is the middle value, so to rewrite the list in ascending order as given below:

13, 13, 13, 13, 14, 14, 16, 18, 21

There are nine numbers in the list, so the middle one will be

9+12=102=5

= 5th number

Hence, the median is 14.

The mode is the number that is repeated more often than any other, so 13 is the mode.

The largest value in the list is 21, and the smallest is 13, so the range is 21 – 13 = 8.

Mean = 15

Median = 14

Mode = 13

Range = 8

Practice questions

Q1) The following table represents the distribution of the annual number of days over 100 degrees Fahrenheit for Dallas-Fort Worth for a sample of 80 years from 1905 to 2004. Find sample standard deviation?

Days Above 100 Degrees

Number of Years

0 - 9

25

10 - 19

33

20 - 29

14

30 - 39

5

40 - 49

2

50 - 59

1

Q2) What is the mean of the following numbers? 10, 39, 71, 39, 76, 38, 25.

Q3) What is the median of the following numbers? 10, 39, 71, 42, 39, 76, 38, 25

Q4) The number of service upgrades sold by each of 30 employees is as follows: 32, 6, 21, 10, 8, 11, 12, 36, 17, 16, 15, 18, 40, 24, 21, 23, 24, 24, 29, 16, 32, 31, 10, 30, 35, 32, 18, 39, 12, 20 What is the median number of service upgrades sold by the 30 employees?

Q5) What is the mode of the following numbers? 12, 11, 14, 10, 8, 13, 11, 9

Q6) The following data represents the age distribution of a sample of 100 people covered by health insurance (private or government). The sample was taken in 2003. Find population standard deviation.

Age

Number

25 - 34

23

35 - 44

29

45 - 54

28

55 - 64

20

Q7)The following data represent the high temperature distribution in degrees Fahrenheit for a sample of 40 days from the month of August in Chicago since 1872. Find sample standard deviation.

Temperature

Days

60 - 69

3

70 - 79

15

80 - 89

17

90 - 99

5

Q8)A sample of college students was asked how much they spent monthly on a cell phone plan (to the nearest dollar). Find sample standard deviation.

Monthly Cell Phone Plan Cost ($)

Number of Students

10 - 19

8

20 - 29

16

30 - 39

21

40 - 49

11

50 - 59

4

Q9) The following data represent the difference in scores between the winning and losing teams in a sample of 15 college football bowl games from XXXXXXXXXXFind sample standard deviation.

Point Difference

Number of Bowl Games

1 - 5

8

6 - 10

0

11 - 15

2

16 - 20

3

21 - 25

1

26 - 30

0

31 - 35

1

Q10) Draw stem and leaf plot of:

23,24,25,56,34,66,47,48,49,28, 29, 69,33

Example 1 – Standard deviation

A hen lays eight eggs. Each egg was weighed and recorded as follows:

60 g, 56 g, 61 g, 68 g, 51 g, 53 g, 69 g, 54 g.

a. First, calculate the mean:

b. Now, find the standard deviation.

Table 1. Weight of eggs, in grams

Weight (x)

(x - )

(x - )2

60

1

1

56

-3

9

61

2

4

68

9

81

51

-8

64

53

-6

36

69

10

100

54

-5

25

472

320

c.

Using the information from the above table, we can see that

In order to calculate the standard deviation, we must use the following formula:

Answered Same DayMay 14, 2021

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