Basic Algebra Concepts: Lesson 1.4

Table of Contents

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:

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

Key Formula for Percentage Problems

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

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

Algebraically:

$P = \frac{c}{100} \cdot W$

Where:

$P$ : The part (portion of the whole).
$W$ : The whole (total quantity).
$c$ : The percent value.

Key Words in Percentage Problems

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

Fractions and Their Equivalent Percents

Here are common fractions and their percentage equivalents:

Fraction	Percentage
$\frac{1}{3}$	$33 \frac{1}{3}\%$
$\frac{2}{3}$	$66 \frac{2}{3}\%$
$\frac{1}{4}$	$25\%$
$\frac{1}{5}$	$20\%$
$\frac{1}{2}$	$50\%$
$\frac{2}{5}$	$40\%$
$\frac{3}{4}$	$75\%$
$\frac{3}{5}$	$60\%$
$\frac{4}{5}$	$80\%$

Key Point

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

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

Converting Percentages and Solving Different Types of Percentage Problems

Example 1: Converting Fractions to Percentages

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

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 $27\%$ to a decimal:

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

Three Types of Percentage Problems

Finding the Part ( $P$ )
Example:
What is 25% of 80?
Solution:
Use the formula:
$P = \frac{c}{100} \cdot W$
Substituting $c = 25$ and $W = 80$ :
$P = \frac{25}{100} \cdot 80 = 20$
Answer: $25\% \text{ of } 80 \text{ is } 20.$
Finding the Whole ( $W$ )
Example:
50 is 40% of what number?
Solution:
Use the formula:
$P = \frac{c}{100} \cdot W$
Rearrange to solve for $W$ :
$W = \frac{P \cdot 100}{c}$
Substituting $P = 50$ and $c = 40$ :
$W = \frac{50 \cdot 100}{40} = 125$
Answer: $50 \text{ is } 40\% \text{ of } 125.$
Finding the Percent ( $c$ )
Example:
If $P = 25$ and $W = 100$ , find $c$ .
Solution:
Use the formula:
$c = \frac{P \cdot 100}{W}$
Substituting $P = 25$ and $W = 100$ :
$c = \frac{25 \cdot 100}{100} = 25\%$
Answer: $25 \text{ is } 25\% \text{ of } 100.$

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:

$P = \frac{c}{100} \cdot W$

Rearranging for $c$ :

$c = \frac{P \cdot 100}{W}$

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

$c = \frac{33 \cdot 100}{165} = 20$

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

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:

Translate the problem into an algebraic expression:
- $\text{Initial total} = 100$
- $25\% \text{ more people come in: } 100 + \frac{25}{100} \cdot 100 = 125$
- $20\% \text{ of the new total leave: } 125 – \frac{20}{100} \cdot 125$
Simplify:
$125 – \frac{20 \cdot 125}{100} = 125 – 25 = 100$

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:

With words: three to two
With ratio signs: 3:2
With fractions: 3/2
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.
$\text{Ratio} = \frac{\text{12 minutes}}{\text{1 hour}}$
Step 2: Convert “1 hour” into minutes.
$\frac{12 \text{ minutes}}{1 \text{ hour}} \times \frac{1 \text{ hour}}{60 \text{ minutes}}$
Step 3: Simplify the fraction.
$= \frac{12}{60}$
Step 4: Reduce to the lowest terms.
$= \frac{1}{5}$
Step 5: Represent the ratio.
$= 1:5$

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 = $3 + 2 = 5$ parts

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

\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:

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

Solve for $x$ :

x = \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., $3 + 2 = 5$ ) to find the total number of parts.
Never use only one part (e.g., $3$ or $2$ ) as the denominator when calculating percentages.

Proportion

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

$\frac{2}{3} = \frac{8}{12}$

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

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

Step 1: Cross-Multiply

$12 \cdot x = 8 \cdot 21$

Step 2: Simplify

$12x = 168$

Step 3: Solve for $x$

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

Step 4: Final Answer

$x = 14$

Final Result:

The value of $x$ is 14.

Practice Problems with Solutions

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: $15 \times 12 + 8 = 188$ inches
- Width: $9 \times 12 + 8 = 116$ inches
  Ratio = $\frac{188}{116} = \frac{47}{29}$ .
  Answer: $47:29$
What is the ratio of 3 pounds to 6 ounces?
Convert pounds to ounces: $3 \text{ lbs } = 3 \times 16 = 48 \text{ oz }$ .
Ratio = $\frac{48}{6} = 8:1$ .
Answer: $8:1$
The ratio of two numbers is 3 to 5. If the larger number is 165, what is the smaller?
Let the smaller number be $x$ :
$\frac{x}{165} = \frac{3}{5}$ .
Cross-multiply: $5x = 3 \times 165$ .
$x = \frac{495}{5} = 99$ .
Answer: 99
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 = $124 + 176 = 300$ .
Ratio = $\frac{176}{300} = \frac{88}{150} = \frac{44}{75}$ .
Answer: $44:75$
If $x$ baseballs cost $d$ dollars, how much will $y$ baseballs cost?
Cost per baseball = $\frac{d}{x}$ .
Cost of $y$ baseballs = $y \cdot \frac{d}{x} = \frac{yd}{x}$ .
Answer: $\frac{yd}{x}$
If 8 men can paint a house in 12 hours, how long would it take 6 men to paint the same house?
Total work = $8 \times 12 = 96$ man-hours.
Time for 6 men = $\frac{96}{6} = 16$ hours.
Answer: 16 hours
If $\frac{19}{5x} = \frac{9}{25}$ , solve for $x$ :
Cross-multiply: $19 \times 25 = 9 \times 5x$
$475 = 45x$
$x = \frac{475}{45} = \frac{95}{9}$
Answer: $\frac{95}{9}$
On a certain map, 1 inch equals 32 miles. How many miles would 5.2 inches equal?
$5.2 \times 32 = 166.4$ miles.
Answer: 166.4 miles
If 3 teaspoons equal 1 tablespoon and 2 tablespoons equal 1 ounce, how many ounces are there in 30 teaspoons?
Convert teaspoons to tablespoons: $\frac{30}{3} = 10$ tablespoons.
Convert tablespoons to ounces: $\frac{10}{2} = 5$ ounces.
Answer: 5 ounces
If snow is falling at the rate of $\frac{1}{3}$ inch per 24 minutes, how much snow will fall in 2 hours?
2 hours = $2 \times 60 = 120$ minutes.
Snowfall in 120 minutes = $\frac{1}{3} \times \frac{120}{24} = \frac{1}{3} \times 5 = \frac{5}{3} = 1.67$ inches.
Answer: $1.67$ 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 $x$ inches farther behind runner A every $y$ minutes. At this rate, how far in feet will runner B be behind runner A after 1 hour?

Options:
- (A) $12xy$
- (B) $\frac{x}{12y}$
- (C) $\frac{12y}{x}$
- (D) $\frac{y}{5x}$
- (E) $\frac{5x}{y}$

5. The difference between $7 \frac{3}{4}$ feet and $5 \frac{5}{6}$ feet in inches is?

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

6. Simplify $\frac{1}{\frac{1}{a} – \frac{1}{b}}$ where $a \neq 0, b \neq 0, a \neq b$

Options:
- (A) $\frac{ab}{a-b}$
- (B) $\frac{a-b}{ab}$
- (C) $b-a$
- (D) $\frac{ab}{b-a}$
- (E) $\frac{b-a}{ab}$

7. 8 × 0.125 = ?

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

8. If $0.2^2 = \sqrt{x}$ , then $x = ?$

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

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

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

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

Options:
- (A) $\frac{1}{3}$
- (B) $-3^2$
- (C) $\frac{3}{10}$
- (D) $\frac{1}{0.3}$
- (E) None of the above

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

Options:
- (A) $2x + 1$
- (B) $2x + 4$
- (C) $\frac{1}{2}x + 2$
- (D) $\frac{x + 4}{4}$
- (E) $\frac{2x + 4}{4}$

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 $A$ and another number is $P$ . The other number must be:

Options:
- (A) $P – A$
- (B) $\frac{PA}{2}$
- (C) $2A – P$
- (D) $2P – A$
- (E) $\frac{2P + A}{2}$

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

Options:
- (A) $\frac{x + 84}{2}$
- (B) $\frac{x + 420}{5}$
- (C) $\frac{10x + 84}{15}$
- (D) $\frac{10x + 420}{15}$
- (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 $6y$ dollars to fence three sides of a square field. How much will it cost to fence the fourth side?

Options:
- (A) $\frac{3y}{4}$
- (B) $y$
- (C) $\frac{4y}{3}$
- (D) $2y$
- (E) $24y$

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

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) $11:1$
- (B) $11:25$
- (C) $11:14$
- (D) $14:11$
- (E) $25:14$

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) $\frac{1}{6}$
- (B) $\frac{1}{3}$
- (C) 1
- (D) $\frac{3}{1}$
- (E) $\frac{60}{1}$

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

One half of the socks in a drawer are brown, $\frac{1}{4}$ of them are black, and $\frac{1}{5}$ of them are blue. If the rest of them are white, what fractional part of the socks are white?
If a $\frac{1}{4}$ inch piece of ribbon costs a nickel, then 1 foot of ribbon costs how much in dollars?
How much more is $\frac{1}{2}$ of $\frac{2}{3}$ than $\frac{3}{4}$ of $\frac{1}{3}$ ?
Jim paints $\frac{1}{3}$ of a fence, Joan paints $\frac{1}{2}$ of what is left. What fraction of the fence is left unpainted?
How many $\frac{1}{10}$ inch pieces of string can be cut from a 16.3-inch string?
Frank can cut a lawn in $2 \frac{1}{2}$ hours; Tom cuts the same lawn in $1 \frac{3}{4}$ hours. What is the average length of time it takes, in hours, if they cut the lawn together?
30% of 80 is what percent of 24?
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?
In a scale drawing, 3 inches represents 9 feet. How many inches represents 1 foot 6 inches? (1 foot = 12 inches)
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 = $\frac{\text{Distance}}{\text{Time}}$
Distance = $\text{Speed} \times \text{Time}$
Time = $\frac{\text{Distance}}{\text{Speed}}$

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:

If a car travels at a rate of 30 mph for $5 \frac{1}{2}$ hours, how many miles will the car travel?
Solution:
- $R = 30 mph, T = 5 \frac{1}{2}$
- Use the formula $R \cdot T = D$
- Calculation: $D = (30)(5 \frac{1}{2}) \\ D = (30)(\frac{11}{2}) \\ D = \frac{330}{2} \\ D = 165 \, \text{miles}.$
- Answer: 165 miles.

If a plane travels 720 miles in $2 \frac{1}{4}$ hours, what is its average speed?
Solution:
- $D = 720 miles, T = 2 \frac{1}{4}$
- Use the formula $R = \frac{D}{T}$
- Calculation: $R = \frac{720}{2 \frac{1}{4}} \\ R = \frac{720}{\frac{9}{4}} \\ R = 720 \cdot \frac{4}{9} \\ R = 320 \, \text{mph}.$
- 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 ( $D$ ) = 420 miles
- Speed ( $R$ ) = 48 mph
- Solve for Time ( $T$ ).
Formula:
$T = \frac{D}{R}$
Calculation:
$T = \frac{420}{48}$ $T = \frac{35}{4}$ $T = 8 \frac{3}{4} \, \text{hours}.$

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 ( $R_1$ ) = 60 mph.
- Car 2 travels at a speed ( $R_2$ ) = 45 mph.
- Since the cars are traveling in opposite directions, the total distance covered by both cars combined in the same time ( $T$ ) is given by: $d_1 + d_2 = 455$
- Expressing this using their rates and time: $60T + 45T = 455$
Step 2: Simplify the equation.
$105T = 455$
Solve for $T$ :
$T = \frac{455}{105} = \frac{13}{3} = 4 \frac{1}{3} \, \text{hours.}$
Step 3: Convert $T = 4 \frac{1}{3} \, \text{hours}$ into hours and 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.
- $1:00 \, \text{PM} + 4 \, \text{hours and 20 minutes} = 5:20 \, \text{PM.}$

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

Motion Problems Practice:

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?
At what speed must a car travel in order to go 940 miles in $15 \frac{2}{3}$ hours?
What is the rate of a boat that travels $pq$ kilometers in $p$ hours?
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?
A boy can pedal his bicycle $\frac{3}{4}$ miles in 6 minutes. What is his rate in mph?
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?
A plane travels $x$ miles during the first 2 hours of a trip and $y$ miles during the last 3 hours of the trip. What was the average rate for the entire trip?
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?
How many minutes would it take for a fire engine to get to a fire $x$ miles away if it travels at a rate of $y$ mph?
How much further can a boat traveling at a rate of 15 kilometers per hour for $x$ hours travel than a second boat traveling at a rate of 18 kilometers per hour for $x-3$ hours?

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. = $8:00 – 1:45 = 6.25 \, \text{hours (6 hours and 15 minutes)}$
Using the formula $D = R \cdot T$ :
$D = 120 \cdot 6.25 = 750 \, \text{miles}$
Answer: 750 miles.

At what speed must a car travel in order to go 940 miles in $15 \frac{2}{3}$ hours?
Solution:
Convert $15 \frac{2}{3}$ to a decimal: $15 \frac{2}{3} = 15.6667$
Using the formula $R = D / T$
$R = 940 / 15.6667 \approx 60 \, \text{mph}$
Answer: 60 mph.

What is the rate of a boat that travels $pq$ kilometers in $p$ hours?
Solution:
Using the formula $R = D / T$ :
$R = \frac{pq}{p} = q \, \text{km/hr}$
Answer: $q \, \text{km/hr}$

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. = $2.5 \, \text{hours}$ .
Time remaining after the stop (from 10:30 A.M. to 12:00 P.M.) = $1.5 \, \text{hours}$
Using $R = D / T$
$R = 105 / 1.5 = 70 \, \text{mph}$
Answer: 70 mph.

A boy can pedal his bicycle $\frac{3}{4}$ miles in 6 minutes. What is his rate in mph?
Solution:
Convert 6 minutes to hours: $6 \, \text{minutes} = \frac{6}{60} = 0.1 \, \text{hours}$ .
Using $R = D / T$ :
$R = 0.75 / 0.1 = 7.5 \, \text{mph}$
Answer: 7.5 mph.

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 = $5 \cdot 2 + 12 \cdot 3 = 10 + 36 = 46 \, \text{miles}$
Total time = $2 + 3 = 5 \, \text{hours}$
Average rate $R = D / T = 46 / 5 = 9.2 \, \text{mph}$
Answer: 9.2 mph.

A plane travels $x$ miles during the first 2 hours of a trip and $y$ miles during the last 3 hours of the trip. What was the average rate for the entire trip?
Solution:
Total distance = $x + y \, \text{miles}$
Total time = $2 + 3 = 5 \, \text{hours}$
Average rate $R = D / T = (x + y) / 5$
Answer: $\frac{x + y}{5} \, \text{mph}$

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 = $40 + 54 = 94 \, \text{mph}$
Using $T = D / R$ :
$T = 188 / 94 = 2 \, \text{hours}$
Answer: 2 hours.

How many minutes would it take for a fire engine to get to a fire $x$ miles away if it travels at a rate of $y$ mph?
Solution:
Using $T = D / R$ :
Time in hours $T = x / y$ .
Convert hours to minutes: $T \cdot 60 = (x / y) \cdot 60 = \frac{60x}{y} \, \text{minutes}$
Answer: $\frac{60x}{y} \, \text{minutes}$

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

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

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

Let me know if you need further clarifications!

Lesson 1-4: Understanding Percentages

What is a Percent?

Key Formula for Percentage Problems

Key Words in Percentage Problems

Fractions and Their Equivalent Percents

Key Point

Converting Percentages and Solving Different Types of Percentage Problems

Example 1: Converting Fractions to Percentages

Key Point for Decimal to Percent Conversion

Example 2: Converting Percent to Decimal

Three Types of Percentage Problems

Solving Percentage Problems with Examples

Example 1: Finding What Percent of One Number Is Another

Example 2: Percentage Increase and Decrease

Key Insights:

Ratio & Proportion

Definition:

Key Tip:

Example Problem:

Key Point:

Example Problem: Ratio and Percentage

Key Note:

Proportion

Final Result:

Practice Problems with Solutions

Questions and Answers

1. 1 – 1/(1 – 1/2) = ?

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

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?

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

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

6. Simplify 11a−1b\frac{1}{\frac{1}{a} – \frac{1}{b}} ​−b1​1​ where a≠0,b≠0,a≠ba \neq 0, b \neq 0, a \neq b

7. 8 × 0.125 = ?

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

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 = ?

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

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

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?

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

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?

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

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?

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?

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?

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?

20. It costs 6y6y6y dollars to fence three sides of a square field. How much will it cost to fence the fourth side?

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

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

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?

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

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

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

Motion Problems – Lesson 2-1

Formula Basics:

Units:

Examples:

Example III: Travel Time Calculation

Example: Cars Traveling in Opposite Directions

Motion Problems Practice: