Ratios & Proportional Relationships

Proportional Relationships and Percent Problems

Grade 7

✓By the end of this lesson students will be able to identify proportional relationships from tables, graphs, and equations.
✓By the end of this lesson students will be able to determine the constant of proportionality (k) in proportional relationships.
✓By the end of this lesson students will be able to represent proportional relationships by equations.
✓By the end of this lesson students will be able to solve real-world problems involving proportional relationships, including those with percents.
✓By the end of this lesson students will be able to calculate simple interest, sales tax, and total cost.

Key concepts

Proportional Relationship

A proportional relationship exists between two quantities when the ratio of one quantity to the other is constant. This means that as one quantity changes, the other quantity changes by a constant factor. A proportional relationship can be represented by an equation of the form y = kx, where k is the constant of proportionality. The graph of a proportional relationship is a straight line that passes through the origin (0,0).

y = kx

Constant of Proportionality (k)

The constant of proportionality, denoted by 'k', is the constant ratio between two proportional quantities, y and x. It represents the unit rate of the relationship. To find the constant of proportionality, you divide the y-value by the corresponding x-value (k = y/x).

k = y/x

Percent

A percent is a ratio that compares a number to 100. The word 'percent' means 'per one hundred'. Percents can be written as fractions with a denominator of 100 or as decimals. For example, 25% can be written as 25/100 or 0.25. When performing calculations with percents, it is essential to convert them to their decimal or fractional form first.

Simple Interest

Simple interest is the amount of money paid or earned for the use of money. It is calculated only on the original principal amount. The formula for simple interest involves the principal (the initial amount of money), the annual interest rate (expressed as a decimal), and the time (in years).

I = Prt (Interest = Principal × Rate × Time)

Sales Tax

Sales tax is a percentage of the cost of goods or services that is added to the price and collected by the government. To calculate the sales tax amount, you multiply the original price by the sales tax rate (expressed as a decimal). The total cost is then the original price plus the sales tax amount.

Sales Tax Amount = Original Price × Sales Tax Rate (as a decimal)

Key facts to remember

1A proportional relationship can be represented by the equation y = kx, where k is the constant of proportionality.
2The constant of proportionality (k) is the unit rate and is found by dividing the y-value by the x-value (k = y/x).
3The graph of a proportional relationship is a straight line that passes through the origin (0,0).
4To convert a percent to a decimal, divide by 100 or move the decimal point two places to the left.
5Simple interest is calculated using the formula I = Prt, where P is the principal, r is the annual interest rate (as a decimal), and t is the time in years.
6Sales tax is a percentage of the original price that is added to find the total cost.

Worked examples

Example 1

The table shows the number of miles a car travels over a certain number of hours at a constant speed. Is this a proportional relationship? If so, find the constant of proportionality and write an equation representing the relationship.

I1. To determine if it's a proportional relationship, check if the ratio of Distance (y) to Time (x) is constant for each pair of values (k = y/x).

II For (2 hours, 120 miles): k = 120 / 2 = 60

III For (3 hours, 180 miles): k = 180 / 3 = 60

IV For (5 hours, 300 miles): k = 300 / 5 = 60

V2. Since the ratio (k) is constant for all pairs (k=60), this is a proportional relationship.

VI3. The constant of proportionality is 60 miles per hour.

VII4. Write the equation in the form y = kx, where y is distance and x is time.

Answer

Yes, it is a proportional relationship. The constant of proportionality is 60. The equation is y = 60x.

The constant of proportionality represents the unit rate, which in this case is the speed of the car in miles per hour.

Example 2

Michael deposits $800 into a savings account that earns 2.5% simple interest per year. How much interest will he earn in 3 years? What will be his total balance after 3 years?

I1. Identify the given values:

II Principal (P) = $800

III Annual Interest Rate (r) = 2.5%

IV Time (t) = 3 years

V2. Convert the annual interest rate from a percentage to a decimal:

VI r = 2.5% = 2.5 / 100 = 0.025

VII3. Use the simple interest formula I = Prt to calculate the interest earned:

VIII I = $800 × 0.025 × 3

9 I = $20 × 3

10 I = $60

114. Calculate the total balance by adding the interest earned to the principal:

12 Total Balance = Principal + Interest

13 Total Balance =

800 +

14 Total Balance = $860

Answer

Michael will earn $60 in interest. His total balance after 3 years will be$ 860.

Remember that the time 't' in the simple interest formula must always be in years.

Example 3

A new video game costs $59.99. The sales tax rate in the state is 6%. What is the sales tax amount? What is the total cost of the video game?

I1. Identify the given values:

II Original Price = $59.99

III Sales Tax Rate = 6%

IV2. Convert the sales tax rate from a percentage to a decimal:

V Sales Tax Rate = 6% = 6 / 100 = 0.06

VI3. Calculate the sales tax amount:

VII Sales Tax Amount = Original Price × Sales Tax Rate

VIII Sales Tax Amount = $59.99 × 0.06

9 Sales Tax Amount = $3.5994

10 Round to the nearest cent: $3.60

114. Calculate the total cost by adding the sales tax amount to the original price:

12 Total Cost = Original Price + Sales Tax Amount

13 Total Cost =

59.99 +

3.60

14 Total Cost = $63.59

Answer

The sales tax amount is $3.60. The total cost of the video game is$ 63.59.

Always round monetary amounts to two decimal places (the nearest cent) at the end of your calculations.

Common mistakes

✗Confusing proportional relationships with non-proportional linear relationships (e.g., y = x + 5, which does not pass through the origin).
✗Not converting the percentage rate to a decimal or fraction before calculating interest or tax amounts.
✗Using time in months or days instead of years for the simple interest formula (I=Prt) without converting it to years.
✗Incorrectly identifying which quantity is x and which is y when finding the constant of proportionality (always k = y/x).
✗Forgetting to add the calculated tax or interest to the original amount to find the total cost or total balance.

Exam tips

★Always check if a relationship's graph passes through the origin (0,0) to confirm if it represents a proportional relationship.
★When working with percents, convert them to decimals or fractions immediately to avoid calculation errors.
★Read word problems carefully to identify the principal, rate, and time for interest problems, or the original price and tax rate for tax problems.
★Show all your steps, especially when converting percents and using formulas, to earn partial credit even if the final answer is incorrect.

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