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Basic Algebra Concepts: Lesson 1.4

Basic Algebra Concepts: Lesson 1

Lesson 1-4: Understanding Percentages

What is a Percent?

  • A percent is essentially a fraction with a denominator of 100.
  • It means “divided by 100”.

Example:
Convert 60% to a fraction:

60%=60100=35(Reduce to lowest terms.)60\% = \frac{60}{100} = \frac{3}{5} \quad (\text{Reduce to lowest terms.}) 

Key Formula for Percentage Problems

Almost every percentage problem can be reduced to the following statement:

“The part is some percent of the whole.”\text{“The part is some percent of the whole.”} 

Algebraically:

P=c100⋅WP = \frac{c}{100} \cdot W 

Where:

  • PP : The part (portion of the whole).
  • WW : The whole (total quantity).
  • cc : The percent value.

Key Words in Percentage Problems

  • Is/are: Represented by an equal sign == .
  • Of: Represents multiplication ⋅\cdot.
  • What percent?: Represented as x100\frac{x}{100}.

Fractions and Their Equivalent Percents

Here are common fractions and their percentage equivalents:

FractionPercentage
13\frac{1}{3}3313%33 \frac{1}{3}\% 
23\frac{2}{3}6623%66 \frac{2}{3}\% 
14\frac{1}{4}25%25\% 
15\frac{1}{5}20%20\% 
12\frac{1}{2}50%50\% 
25\frac{2}{5}40%40\% 
34\frac{3}{4}75%75\% 
35\frac{3}{5}60%60\% 
45\frac{4}{5}80%80\% 

Key Point

To convert a fraction to a percent, follow these steps:

  1. Divide the numerator by the denominator.
  2. Move the decimal point two places to the right.

Converting Percentages and Solving Different Types of Percentage Problems

Example 1: Converting Fractions to Percentages

  1. Convert the fraction 35\frac{3}{5}  to a decimal: 35=0.6(Divide numerator by denominator.)\frac{3}{5} = 0.6 \quad (\text{Divide numerator by denominator.}) 
  2. Convert the decimal to a percentage: 0.6=60%(Move the decimal point two places to the right.)0.6 = 60\% \quad (\text{Move the decimal point two places to the right.}) 

Key Point for Decimal to Percent Conversion

To convert a decimal to a percent, move the decimal point two places to the right.

Example 2: Converting Percent to Decimal

Convert 27%27\% to a decimal:

27%=0.27(Move the decimal point two places to the left.)27\% = 0.27 \quad (\text{Move the decimal point two places to the left.}) 

Three Types of Percentage Problems

  1. Finding the Part (PP )

    Example:
    What is 25% of 80?
    Solution:
    Use the formula:

    P=c100⋅WP = \frac{c}{100} \cdot W 

    Substituting c=25c = 25  and W=80W = 80 :

    P=25100⋅80=20P = \frac{25}{100} \cdot 80 = 20 

    Answer: 25% of 80 is 20.25\% \text{ of } 80 \text{ is } 20. 

  2. Finding the Whole (WW )

    Example:
    50 is 40% of what number?
    Solution:
    Use the formula:

    P=c100⋅WP = \frac{c}{100} \cdot W 

    Rearrange to solve for WW :

    W=P⋅100cW = \frac{P \cdot 100}{c}

    Substituting P=50P = 50  and c=40c = 40 :

    W=50⋅10040=125W = \frac{50 \cdot 100}{40} = 125 

    Answer: 50 is 40% of 125.50 \text{ is } 40\% \text{ of } 125. 

  3. Finding the Percent (cc )

    Example:
    If P=25P = 25  and W=100W = 100 , find cc .
    Solution:
    Use the formula:

    c=P⋅100Wc = \frac{P \cdot 100}{W}

    Substituting P=25P = 25  and W=100W = 100 :

    c=25⋅100100=25%c = \frac{25 \cdot 100}{100} = 25\% 

    Answer: 25 is 25% of 100.25 \text{ is } 25\% \text{ of } 100. 

Solving Percentage Problems with Examples

Example 1: Finding What Percent of One Number Is Another

Question:
What percent of 165 is 33?

Solution:
Use the formula:

P=c100⋅WP = \frac{c}{100} \cdot W 

Rearranging for cc :

c=P⋅100Wc = \frac{P \cdot 100}{W}

Substitute P=33P = 33  and W=165W = 165 :

c=33⋅100165=20c = \frac{33 \cdot 100}{165} = 20 

Answer: 33 is 20% of 165.33 \text{ is } 20\% \text{ of } 165. 

Example 2: Percentage Increase and Decrease

Question:
There are 100 people in a football stadium. If 25% more people come in and then 20% of the new total leave, how many people remain?

Solution:

  1. Translate the problem into an algebraic expression:

    • Initial total=100\text{Initial total} = 100
    • 25% more people come in: 100+25100⋅100=12525\% \text{ more people come in: } 100 + \frac{25}{100} \cdot 100 = 125 
    • 20% of the new total leave: 125−20100⋅12520\% \text{ of the new total leave: } 125 – \frac{20}{100} \cdot 125 

  2. Simplify:

    125−20⋅125100=125−25=100125 – \frac{20 \cdot 125}{100} = 125 – 25 = 100 

Answer: 100 people remain.

Key Insights:

  • For percentage increases, add the additional percent to the total before applying further operations.
  • For percentage decreases, subtract the specified percentage of the total from the total.
  • Pay attention to the phrasing in problems (e.g., “how many more” versus “how many total”).

 

Ratio & Proportion

Definition:

A ratio is a comparison between the number of elements in one group and the number of elements in another group. Ratios can be represented in multiple ways:

  1. With words: three to two
  2. With ratio signs: 3:2
  3. With fractions: 3/2
  4. With percents: 1.50%

Key Tip:

In most problems, converting the ratio to its fractional form makes it easier to work with.

Example Problem:

What is the ratio of 12 minutes to 1 hour?

  • Step 1: Express the ratio as a fraction.

    Ratio=12 minutes1 hour\text{Ratio} = \frac{\text{12 minutes}}{\text{1 hour}}

  • Step 2: Convert “1 hour” into minutes.

    12 minutes1 hour×1 hour60 minutes\frac{12 \text{ minutes}}{1 \text{ hour}} \times \frac{1 \text{ hour}}{60 \text{ minutes}}

  • Step 3: Simplify the fraction.

    =1260= \frac{12}{60}

  • Step 4: Reduce to the lowest terms.

    =15= \frac{1}{5}

  • Step 5: Represent the ratio.

    =1:5= 1:5 

Key Point:

  • Ratios compare different parts of a group to each other.
  • Fractions compare a part to the whole.

Example Problem: Ratio and Percentage

Question:
In a math class, the ratio of the number of boys to the number of girls is 3:2. What percent of the class consists of girls?

Step 1: Identify the total parts in the ratio.
The ratio 3:2 means:

  • Boys = 3 parts
  • Girls = 2 parts
  • Total class = 3+2=53 + 2 = 5  parts

Step 2: Find the fraction of the class that is girls.

Fraction of girls=Girls’ partsTotal parts=25\text{Fraction of girls} = \frac{\text{Girls’ parts}}{\text{Total parts}} = \frac{2}{5}

Step 3: Convert the fraction to a percent.
Set up the proportion:

25=x100\frac{2}{5} = \frac{x}{100}

Solve for xx :

x=2×1005=40x = \frac{2 \times 100}{5} = 40 

Final Answer:
The percentage of the class that consists of girls is 40%.

Key Note:

  • Always add the numbers in the ratio (e.g., 3+2=53 + 2 = 5 ) to find the total number of parts.
  • Never use only one part (e.g., 33  or 22 ) as the denominator when calculating percentages.
 

Proportion

Definition:
A proportion is two ratios set equal to each other.
Example:

23=812\frac{2}{3} = \frac{8}{12}

An important point about proportions is that the cross-products are equal. This property is often used to solve equations.

Example Problem:
If x8=2112\frac{x}{8} = \frac{21}{12} , solve for xx .

Step 1: Cross-Multiply

12⋅x=8⋅2112 \cdot x = 8 \cdot 21 

Step 2: Simplify

12x=16812x = 168 

Step 3: Solve for xx 

x=16812x = \frac{168}{12}

Step 4: Final Answer

x=14x = 14 

Final Result:

The value of xx is 14.

Practice Problems with Solutions

  1. A room is 15 feet 8 inches long, and 9 feet 8 inches wide. What is the ratio of the length to the width?
    Convert measurements to inches:

    • Length: 15×12+8=18815 \times 12 + 8 = 188  inches
    • Width: 9×12+8=1169 \times 12 + 8 = 116  inches
      Ratio = 188116=4729\frac{188}{116} = \frac{47}{29} .
      Answer: 47:2947:29 

  2. What is the ratio of 3 pounds to 6 ounces?
    Convert pounds to ounces: 3 lbs =3×16=48 oz 3 \text{ lbs } = 3 \times 16 = 48 \text{ oz }.
    Ratio = 486=8:1\frac{48}{6} = 8:1 .
    Answer: 8:18:1 

  3. The ratio of two numbers is 3 to 5. If the larger number is 165, what is the smaller?
    Let the smaller number be xx :
    x165=35\frac{x}{165} = \frac{3}{5} .
    Cross-multiply: 5x=3×1655x = 3 \times 165 .
    x=4955=99x = \frac{495}{5} = 99 .
    Answer: 99

  4. If a school has 124 boys and 176 girls, what is the ratio of girls to the total number of students in the school?
    Total students = 124+176=300124 + 176 = 300 .
    Ratio = 176300=88150=4475\frac{176}{300} = \frac{88}{150} = \frac{44}{75} .
    Answer: 44:7544:75 

  5. If xx baseballs cost dd dollars, how much will yy baseballs cost?
    Cost per baseball = dx\frac{d}{x}.
    Cost of yy baseballs = y⋅dx=ydxy \cdot \frac{d}{x} = \frac{yd}{x} .
    Answer: ydx\frac{yd}{x}

  6. If 8 men can paint a house in 12 hours, how long would it take 6 men to paint the same house?
    Total work = 8×12=968 \times 12 = 96  man-hours.
    Time for 6 men = 966=16\frac{96}{6} = 16  hours.
    Answer: 16 hours

  7. If 195x=925\frac{19}{5x} = \frac{9}{25} , solve for xx :
    Cross-multiply: 19×25=9×5x19 \times 25 = 9 \times 5x 
    475=45x475 = 45x 
    x=47545=959x = \frac{475}{45} = \frac{95}{9} 
    Answer: 959\frac{95}{9}

  8. On a certain map, 1 inch equals 32 miles. How many miles would 5.2 inches equal?
    5.2×32=166.45.2 \times 32 = 166.4  miles.
    Answer: 166.4 miles

  9. If 3 teaspoons equal 1 tablespoon and 2 tablespoons equal 1 ounce, how many ounces are there in 30 teaspoons?
    Convert teaspoons to tablespoons: 303=10\frac{30}{3} = 10  tablespoons.
    Convert tablespoons to ounces: 102=5\frac{10}{2} = 5  ounces.
    Answer: 5 ounces

  10. If snow is falling at the rate of 13\frac{1}{3}  inch per 24 minutes, how much snow will fall in 2 hours?
    2 hours = 2×60=1202 \times 60 = 120  minutes.
    Snowfall in 120 minutes = 13×12024=13×5=53=1.67\frac{1}{3} \times \frac{120}{24} = \frac{1}{3} \times 5 = \frac{5}{3} = 1.67 inches.
    Answer: 1.671.67 inches

Questions and Answers

1. 1 – 1/(1 – 1/2) = ?

  • Options:
    • (A) -2
    • (B) -1
    • (C) 0
    • (D) 1
    • (E) 2

2. If 5/x = 15/9, then x = ?

  • Options:
    • (A) 1
    • (B) 3
    • (C) 9
    • (D) 18
    • (E) 27

3. How many X cards would have to be taken from pile A and put into pile B for the fractional part of X cards to be the same in both piles?

  • Options:
    • (A) None
    • (B) 1
    • (C) 2
    • (D) 3
    • (E) 4

4. In a race, runner B falls xx inches farther behind runner A every yy minutes. At this rate, how far in feet will runner B be behind runner A after 1 hour?

  • Options:
    • (A) 12xy12xy 
    • (B) x12y\frac{x}{12y}
    • (C) 12yx\frac{12y}{x}
    • (D) y5x\frac{y}{5x}
    • (E) 5xy\frac{5x}{y}

5. The difference between 7347 \frac{3}{4}  feet and 5565 \frac{5}{6} feet in inches is?

  • Options:
    • (A) 12
    • (B) 12.5
    • (C) 18
    • (D) 23
    • (E) 25

6. Simplify 11a−1b\frac{1}{\frac{1}{a} – \frac{1}{b}} where a≠0,b≠0,a≠ba \neq 0, b \neq 0, a \neq b 

  • Options:
    • (A) aba−b\frac{ab}{a-b}
    • (B) a−bab\frac{a-b}{ab}
    • (C) b−ab-a 
    • (D) abb−a\frac{ab}{b-a}
    • (E) b−aab\frac{b-a}{ab}

7. 8 × 0.125 = ?

  • Options:
    • (A) 0
    • (B) 1
    • (C) 0.1
    • (D) 8.125
    • (E) 0.825

8. If 0.22=x0.2^2 = \sqrt{x}, then x=?x = ? 

  • Options:
    • (A) 0.2
    • (B) 0.02
    • (C) 0.04
    • (D) 0.016
    • (E) 0.0016

9. If −0.6(0.4−p)=1.2(0.8p+0.7p)-0.6(0.4 – p) = 1.2(0.8p + 0.7p) , then p=?p = ? 

  • Options:
    • (A) -5
    • (B) -0.2
    • (C) 0.2
    • (D) 0.5
    • (E) None of the above

10. If 10N=3.33333…10N = 3.33333…  and N=0.33333…N = 0.33333… , then NN  can be rewritten as:

  • Options:
    • (A) 13\frac{1}{3}
    • (B) −32-3^2 
    • (C) 310\frac{3}{10}
    • (D) 10.3\frac{1}{0.3}
    • (E) None of the above

 

11. What is the average of 2x+1,x+5,1−4x,3x+12x + 1, x + 5, 1 – 4x, 3x + 1?

  • Options:
    • (A) 2x+12x + 1 
    • (B) 2x+42x + 4 
    • (C) 12x+2\frac{1}{2}x + 2 
    • (D) x+44\frac{x + 4}{4}
    • (E) 2x+44\frac{2x + 4}{4}

12. Three members of a basketball team have weights that range from 150 to 175 pounds. Which of the following cannot possibly be the average weight of the three players?

  • Options:
    • (A) 160
    • (B) 165
    • (C) 170
    • (D) 155
    • (E) 175

13. The average of AA  and another number is PP . The other number must be:

  • Options:
    • (A) P−AP – A 
    • (B) PA2\frac{PA}{2}
    • (C) 2A−P2A – P 
    • (D) 2P−A2P – A 
    • (E) 2P+A2\frac{2P + A}{2}

14. The average grade of 10 students is xx . If 5 other students each earned a grade of 84, what would be the average grade of the entire group?

  • Options:
    • (A) x+842\frac{x + 84}{2}
    • (B) x+4205\frac{x + 420}{5}
    • (C) 10x+8415\frac{10x + 84}{15}
    • (D) 10x+42015\frac{10x + 420}{15}
    • (E) None of the above

15. After picking 120 peaches, a woman eats 12 of them. What percent remains?

  • Options:
    • (A) 10
    • (B) 30
    • (C) 50
    • (D) 70
    • (E) 90

16. If a boy must walk 12 miles to school and he has completed 75% of the trip, how many miles does he have left to go?

  • Options:
    • (A) 3
    • (B) 4
    • (C) 6
    • (D) 8
    • (E) 9

17. A 60-gallon tank is 40% full of water. If the water is then poured into a 40-gallon tank, what percent of the 40-gallon tank has been filled?

  • Options:
    • (A) 24
    • (B) 40
    • (C) 60
    • (D) 96
    • (E) 100

18. If 30% of a class consists of boys and there are 21 girls in the class, how many boys are there in the class?

  • Options:
    • (A) 30
    • (B) 9
    • (C) 60
    • (D) 42
    • (E) 10

19. After taking 30 socks out of the dryer, Debbie noticed that the ratio of blue socks to brown socks to black socks was 2:3:5. How many black socks were in the dryer?

  • Options:
    • (A) 21
    • (B) 15
    • (C) 9
    • (D) 6
    • (E) 3

20. It costs 6y6y dollars to fence three sides of a square field. How much will it cost to fence the fourth side?

  • Options:
    • (A) 3y4\frac{3y}{4}
    • (B) yy
    • (C) 4y3\frac{4y}{3}
    • (D) 2y2y 
    • (E) 24y24y 

21. If 3:43:4  is equivalent to a:12a:12 , then a=?a = ? 

  • Options:
    • (A) 1
    • (B) 9
    • (C) 11
    • (D) 12
    • (E) 14

22. In a class of 25 students, 44% are boys. What is the ratio of boys to girls in the class?

  • Options:
    • (A) 11:111:1 
    • (B) 11:2511:25 
    • (C) 11:1411:14 
    • (D) 14:1114:11 
    • (E) 25:1425:14 

23. In 15 years, the ratio of my age to my father’s age will be 1:2. Five years ago, the ratio of my age to his was 1:4. How old am I?

  • Options:
    • (A) 10
    • (B) 15
    • (C) 30
    • (D) 40
    • (E) 60

24. A machine can copy 6 pages in 9 seconds. How many pages can it copy in 24 minutes?

  • Options:
    • (A) 12
    • (B) 16
    • (C) 36
    • (D) 960
    • (E) 2,160

25. The ratio of the length of a side of an equilateral triangle to the perimeter of the triangle is:

  • Options:
    • (A) 16\frac{1}{6}
    • (B) 13\frac{1}{3}
    • (C) 1
    • (D) 31\frac{3}{1}
    • (E) 601\frac{60}{1}

For Problems 26–35, enter your solutions into the grids that follow the questions:

  1. One half of the socks in a drawer are brown, 14\frac{1}{4}  of them are black, and 15\frac{1}{5}  of them are blue. If the rest of them are white, what fractional part of the socks are white?

  2. If a 14\frac{1}{4} inch piece of ribbon costs a nickel, then 1 foot of ribbon costs how much in dollars?

  3. How much more is 12\frac{1}{2}  of 23\frac{2}{3} than 34\frac{3}{4}  of 13\frac{1}{3} ?

  4. Jim paints 13\frac{1}{3} of a fence, Joan paints 12\frac{1}{2}  of what is left. What fraction of the fence is left unpainted?

  5. How many 110\frac{1}{10} inch pieces of string can be cut from a 16.3-inch string?

  6. Frank can cut a lawn in 2122 \frac{1}{2} hours; Tom cuts the same lawn in 1341 \frac{3}{4}  hours. What is the average length of time it takes, in hours, if they cut the lawn together?

  7. 30% of 80 is what percent of 24?

  8. If the cost of a 4-minute telephone call is $0.24, then what is the cost in dollars of a 15-minute call at the same rate?

  9. In a scale drawing, 3 inches represents 9 feet. How many inches represents 1 foot 6 inches? (1 foot = 12 inches)

  10. Michele, Ned, and Owen split the award for a contest in the ratio of 6:2:1, respectively. If the total award was worth $72.00, then Ned received how many dollars?

Motion Problems – Lesson 2-1

Formula Basics:

  • Speed = DistanceTime\frac{\text{Distance}}{\text{Time}}
  • Distance = Speed×Time\text{Speed} \times \text{Time} 
  • Time = DistanceSpeed\frac{\text{Distance}}{\text{Speed}}

Units:

  • Speed is measured in: meters/second (m/s), miles/hour (mph), kilometers/hour (km/h).
  • Distance is measured in: meters, miles, kilometers.
  • Time is measured in: hours (hrs), minutes (mins), seconds (secs).

Examples:

  1. If a car travels at a rate of 30 mph for 5125 \frac{1}{2}  hours, how many miles will the car travel?

    Solution:

    • R=30 mph, T=512 
    • Use the formula R⋅T=DR \cdot T = D 
    • Calculation: D=(30)(512)D=(30)(112)D=3302D=165 miles.D = (30)(5 \frac{1}{2}) \\ D = (30)(\frac{11}{2}) \\ D = \frac{330}{2} \\ D = 165 \, \text{miles}. 
    • Answer: 165 miles.

  1. If a plane travels 720 miles in 2142 \frac{1}{4}  hours, what is its average speed?

    Solution:

    • D=720 miles, T=214  
    • Use the formula R=DTR = \frac{D}{T} 
    • Calculation: R=720214R=72094R=720⋅49R=320 mph.R = \frac{720}{2 \frac{1}{4}} \\ R = \frac{720}{\frac{9}{4}} \\ R = 720 \cdot \frac{4}{9} \\ R = 320 \, \text{mph}. 
    • Answer: 320 mph.

 

Example III: Travel Time Calculation

Problem: How long will it take a car to travel 420 miles at an average speed of 48 mph?

Solution:

  • Given:

    • Distance (DD ) = 420 miles
    • Speed (RR ) = 48 mph
    • Solve for Time (TT ).

  • Formula:

    T=DRT = \frac{D}{R}

  • Calculation:

    T=42048T = \frac{420}{48} T=354T = \frac{35}{4} T=834 hours.T = 8 \frac{3}{4} \, \text{hours}. 

Answer: It will take 8 hours and 45 minutes to travel 420 miles at 48 mph.

Example: Cars Traveling in Opposite Directions

Problem: Two cars traveling in opposite directions pass each other at 1:00 PM. One car travels at 60 mph, while the other travels at 45 mph. At what time will the cars be 455 miles apart?

Solution:

  • Step 1: Understand the motion of both cars.

    • Car 1 travels at a speed (R1R_1) = 60 mph.
    • Car 2 travels at a speed (R2R_2) = 45 mph.
    • Since the cars are traveling in opposite directions, the total distance covered by both cars combined in the same time (TT ) is given by: d1+d2=455d_1 + d_2 = 455 
    • Expressing this using their rates and time: 60T+45T=45560T + 45T = 455 

  • Step 2: Simplify the equation.

    105T=455105T = 455 

    Solve for TT :

    T=455105=133=413 hours.T = \frac{455}{105} = \frac{13}{3} = 4 \frac{1}{3} \, \text{hours.} 

  • Step 3: Convert T=413 hoursT = 4 \frac{1}{3} \, \text{hours}  into hours and minutes.

    • 13 hours=1/3×60=20 minutes.\frac{1}{3} \, \text{hours} = 1/3 \times 60 = 20 \, \text{minutes.} 
    • Total time: 4 hours and 20 minutes.

  • Step 4: Add this time to 1:00 PM.

    • 1:00 PM+4 hours and 20 minutes=5:20 PM.1:00 \, \text{PM} + 4 \, \text{hours and 20 minutes} = 5:20 \, \text{PM.}

Answer: The cars will be 455 miles apart at 5:20 PM.

Motion Problems Practice:

  1. What is the distance a plane can travel from 1:45 a.m. to 8:00 a.m. flying at a rate of 120 mph?

  2. At what speed must a car travel in order to go 940 miles in 152315 \frac{2}{3}  hours?

  3. What is the rate of a boat that travels pqpq  kilometers in pp  hours?

  4. Mr. Smith left his house at 7:30 A.M. and drove at a rate of 50 mph until 10:00 A.M. He then stopped for half an hour. At what rate must he travel in order to go to work by noon if he still has 105 miles to go?

  5. A boy can pedal his bicycle 34\frac{3}{4}  miles in 6 minutes. What is his rate in mph?

  6. A boy walks at a rate of 5 mph for 2 hours and then rides his bike at a rate of 12 mph for 3 hours. What is his average rate for the entire trip?

  7. A plane travels xx  miles during the first 2 hours of a trip and yy  miles during the last 3 hours of the trip. What was the average rate for the entire trip?

  8. Two cars travel in opposite directions starting from the same point. One car travels at a rate of 40 mph, and the other car travels at a rate of 54 mph. How long will it take for the two cars to be 188 miles apart?

  9. How many minutes would it take for a fire engine to get to a fire xx miles away if it travels at a rate of yy  mph?

  10. How much further can a boat traveling at a rate of 15 kilometers per hour for xx  hours travel than a second boat traveling at a rate of 18 kilometers per hour for x−3x-3  hours?

 

 

  1. What is the distance a plane can travel from 1:45 a.m. to 8:00 a.m. flying at a rate of 120 mph?

    Solution:
    Time difference from 1:45 a.m. to 8:00 a.m. = 8:00−1:45=6.25 hours (6 hours and 15 minutes)8:00 – 1:45 = 6.25 \, \text{hours (6 hours and 15 minutes)} 
    Using the formula D=R⋅TD = R \cdot T :
    D=120⋅6.25=750 milesD = 120 \cdot 6.25 = 750 \, \text{miles} 

    Answer: 750 miles.

  1. At what speed must a car travel in order to go 940 miles in 152315 \frac{2}{3} hours?

    Solution:
    Convert 152315 \frac{2}{3} to a decimal: 1523=15.666715 \frac{2}{3} = 15.6667 
    Using the formula R=D/TR = D / T 
    R=940/15.6667≈60 mphR = 940 / 15.6667 \approx 60 \, \text{mph} 

    Answer: 60 mph.

  1. What is the rate of a boat that travels pqpq  kilometers in pp  hours?

    Solution:
    Using the formula R=D/TR = D / T :
    R=pqp=q km/hrR = \frac{pq}{p} = q \, \text{km/hr} 

    Answer: q km/hrq \, \text{km/hr} 

  1. Mr. Smith left his house at 7:30 A.M. and drove at a rate of 50 mph until 10:00 A.M. He then stopped for half an hour. At what rate must he travel in order to go to work by noon if he still has 105 miles to go?

    Solution:
    Time from 7:30 A.M. to 10:00 A.M. = 2.5 hours2.5 \, \text{hours}.
    Time remaining after the stop (from 10:30 A.M. to 12:00 P.M.) = 1.5 hours1.5 \, \text{hours} 
    Using R=D/TR = D / T 
    R=105/1.5=70 mphR = 105 / 1.5 = 70 \, \text{mph} 

    Answer: 70 mph.

  1. A boy can pedal his bicycle 34\frac{3}{4} miles in 6 minutes. What is his rate in mph?

    Solution:
    Convert 6 minutes to hours: 6 minutes=660=0.1 hours6 \, \text{minutes} = \frac{6}{60} = 0.1 \, \text{hours}.
    Using R=D/TR = D / T :
    R=0.75/0.1=7.5 mphR = 0.75 / 0.1 = 7.5 \, \text{mph} 

    Answer: 7.5 mph.

  1. A boy walks at a rate of 5 mph for 2 hours and then rides his bike at a rate of 12 mph for 3 hours. What is his average rate for the entire trip?

    Solution:
    Total distance = 5⋅2+12⋅3=10+36=46 miles5 \cdot 2 + 12 \cdot 3 = 10 + 36 = 46 \, \text{miles} 
    Total time = 2+3=5 hours2 + 3 = 5 \, \text{hours} 
    Average rate R=D/T=46/5=9.2 mphR = D / T = 46 / 5 = 9.2 \, \text{mph} 

    Answer: 9.2 mph.

  1. A plane travels xx miles during the first 2 hours of a trip and yy miles during the last 3 hours of the trip. What was the average rate for the entire trip?

    Solution:
    Total distance = x+y milesx + y \, \text{miles} 
    Total time = 2+3=5 hours2 + 3 = 5 \, \text{hours} 
    Average rate R=D/T=(x+y)/5R = D / T = (x + y) / 5 

    Answer: x+y5 mph\frac{x + y}{5} \, \text{mph} 

  1. Two cars travel in opposite directions starting from the same point. One car travels at a rate of 40 mph, and the other car travels at a rate of 54 mph. How long will it take for the two cars to be 188 miles apart?

    Solution:
    Combined rate = 40+54=94 mph40 + 54 = 94 \, \text{mph} 
    Using T=D/RT = D / R :
    T=188/94=2 hoursT = 188 / 94 = 2 \, \text{hours} 

    Answer: 2 hours.

  1. How many minutes would it take for a fire engine to get to a fire xx  miles away if it travels at a rate of yy  mph?

    Solution:
    Using T=D/RT = D / R :
    Time in hours T=x/yT = x / y .
    Convert hours to minutes: T⋅60=(x/y)⋅60=60xy minutesT \cdot 60 = (x / y) \cdot 60 = \frac{60x}{y} \, \text{minutes} 

    Answer: 60xy minutes\frac{60x}{y} \, \text{minutes}

  1. How much further can a boat traveling at a rate of 15 kilometers per hour for xx  hours travel than a second boat traveling at a rate of 18 kilometers per hour for x−3x – 3  hours?

Solution:
Distance of first boat = 15x km15x \, \text{km} 
Distance of second boat = 18(x−3)=18x−54 km18(x – 3) = 18x – 54 \, \text{km} 
Difference = 15x−(18x−54)=−3x+54 km15x – (18x – 54) = -3x + 54 \, \text{km} 

Answer: −3x+54 km-3x + 54 \, \text{km} 

Let me know if you need further clarifications!

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