Business Math Fundamentals: Key Concepts & Calculations

Question 1: Decimal Representation of a Fraction

Question: Consider the fraction 6/7. The decimal representation of this fraction is:

Answer: 6 ÷ 7 = 0.857142857... (repeating)

Question 2: Vaccinated to Unvaccinated Ratio

Question: If 60% of a population is vaccinated, what is the ratio of vaccinated to unvaccinated individuals?

Answer: 60% vaccinated → 60 : 40 → Simplified = 3 : 2

Question 3: Property Tax Calculation

Question: A property has been assessed at $225,000. The mill rate is 14.5. To find the property tax, you would multiply the assessed value by:

Answer: The mill rate of 14.5 means $14.50 per $1,000 of assessed value. To convert this to a decimal factor, divide by 1,000:

• 14.5 ÷ 1,000 = 0.0145
• Property tax = $225,000 × 0.0145 = $3,262.50

Question 4: Sales Commission Earnings

Question: Ben earns 7% on the first $10,000 of sales and 8¾% on sales above that. Total sales = $15,000.

Answer:

• 7% of $10,000 = $700
• Sales above $10,000 = $15,000 - $10,000 = $5,000
• 8.75% (8¾%) of $5,000 = $437.50
• Total commission = $700 + $437.50 = $1,137.50

Question 5: Rounding Decimals

Question: Round 18.9999 to one decimal place.

Answer: 19.0

Question 6: Identifying Fixed and Variable Costs from a Graph

Question: A graph of cost versus number of units shows a straight line. Identify fixed and variable costs.

Answer:

• Fixed cost = y-intercept (the cost when zero units are produced)
• Variable cost = slope of the line (change in cost per unit, calculated as rise/run)

Question 7: Overtime Pay Calculation

Question: Sally worked 49 hours at $16/hour. Overtime is 1.5 times the regular pay rate.

Answer:

• Regular hours pay: 40 hours × $16/hour = $640
• Overtime hours: 49 - 40 = 9 hours
• Overtime rate: 1.5 × $16/hour = $24/hour
• Overtime pay: 9 hours × $24/hour = $216
• Total pay = $640 + $216 = $856

Question 8: Calculating Hourly Rate

Question: Find the hourly rate if weekly pay is $720 and hours worked are 45.

Answer: $720 ÷ 45 hours = $16/hour

Question 9: Weighted Average Cost of a Mix

Question: A seed mix consists of: 10kg @ $2/kg, 5kg @ $3/kg, and 5kg @ $4/kg.

Answer:

• Total cost = (10kg × $2) + (5kg × $3) + (5kg × $4) = $20 + $15 + $20 = $55
• Total quantity = 10kg + 5kg + 5kg = 20kg
• Weighted average cost = $55 ÷ 20kg = $2.75/kg

Question 10: Total Pay with Overtime

Question: Calculate total pay with overtime at 1.5 times the regular rate. 45 hours worked, $16/hour regular rate.

Answer:

• Regular pay (40 hours): 40 × $16 = $640
• Overtime hours: 45 - 40 = 5 hours
• Overtime rate: 1.5 × $16 = $24/hour
• Overtime pay: 5 × $24 = $120
• Total pay = $640 + $120 = $760

Question 11: Salesperson's Total Pay with Commission

Question: A salesperson earns a $950 base salary plus 6% commission on sales over $5,000. If total sales are $6,125, what is the total pay?

Answer:

• Sales eligible for commission = $6,125 - $5,000 = $1,125
• Commission earned = $1,125 × 6% (0.06) = $67.50
• Total pay = Base salary + Commission = $950 + $67.50 = $1,017.50

Question 12: Budget Allocation by Ratio

Question: A store splits a $1,500 budget in a 3:2:1 ratio (Marketing: Operations: Research). How much goes to research?

Answer:

• Total parts in ratio = 3 + 2 + 1 = 6 parts
• Research's share = 1 part out of 6
• Amount for research = (1/6) × $1,500 = $250

Question 13: Loss Coverage by Fractional Ratio

Question: A $180,000 loss is covered by 3 companies in the ratio 2/5 : 1/3 : 1/6. How much does the second company pay?

Answer:

To work with the ratio, convert the fractions to a common denominator (LCM of 5, 3, 6 is 30):

• 2/5 = 12/30
• 1/3 = 10/30
• 1/6 = 5/30

The ratio in whole numbers is 12 : 10 : 5.

• Total parts = 12 + 10 + 5 = 27 parts
• The second company's share = 10 parts out of 27
• Amount paid by second company = (10/27) × $180,000 = $66,666.67 (rounded to two decimal places)

Question 14: Federal Tax Calculation

Question: Calculate federal tax for:

1. $150,000 taxable income
2. $80,000 taxable income

Answer: (Assuming specific tax brackets for demonstration purposes)

i. Tax on $150,000:

• 15% on first $44,701 = $44,701 × 0.15 = $6,705.15
• 22% on next $44,700 (up to $89,401) = $44,700 × 0.22 = $9,834.00
• 26% on next $49,185 (up to $138,586) = $49,185 × 0.26 = $12,788.10
• 29% on remaining $11,414 ($150,000 - $138,586) = $11,414 × 0.29 = $3,309.06
• Total federal tax = $6,705.15 + $9,834.00 + $12,788.10 + $3,309.06 = $32,636.31

ii. Tax on $80,000:

• 15% on first $44,701 = $44,701 × 0.15 = $6,705.15
• 22% on remaining $35,299 ($80,000 - $44,701) = $35,299 × 0.22 = $7,765.78
• Total federal tax = $6,705.15 + $7,765.78 = $14,470.93

Question 15: Percentage Increase in Income

Question: Alex earned $45,000 in 2021 and $48,000 in 2022. What is the percent increase?

Answer:

• Increase in income = $48,000 - $45,000 = $3,000
• Percentage increase = ($3,000 / $45,000) × 100% = 6.67% (rounded to two decimal places)

Question 17: Budget Split by Ratio

Question: A $90,000 budget is split in a 4:5:1 ratio. How much goes to the second category?

Answer:

• Total parts in ratio = 4 + 5 + 1 = 10 parts
• Second category's share = 5 parts out of 10
• Amount for second category = (5/10) × $90,000 = $45,000

Question 18: Percentage Change in Price Index

Question: If the base year price index is 100 and the current index is 110, what's the percentage change?

Answer:

• Percentage Change = ((Current Index - Base Index) / Base Index) × 100%
• Percentage Change = ((110 - 100) / 100) × 100% = (10 / 100) × 100% = 10%

Question 19: Break-Even Analysis and Margin of Safety

Question: A smoothie sells for $6. Cost per unit = $2.50. Fixed costs = $4,000.

Answer:

i. Calculate Break-Even Point:

• Contribution Margin per unit = Selling Price - Variable Cost = $6 - $2.50 = $3.50
• Break-even point (units) = Fixed Costs / Contribution Margin per unit = $4,000 / $3.50 = 1,143 units (rounded up to the nearest whole unit)

ii. Break-Even Point with Reduced Fixed Costs:

• If fixed costs drop to $3,200:
• New Break-even point (units) = $3,200 / $3.50 = 915 units (rounded up to the nearest whole unit)

iii. Calculate Margin of Safety:

(Assuming actual or budgeted sales are 1,200 units)

• Margin of Safety (units) = Actual Sales - Break-even Sales = 1,200 - 1,143 = 57 units
• Margin of Safety (%) = (Margin of Safety in units / Actual Sales in units) × 100%
• Margin of Safety (%) = (57 / 1,200) × 100% = 4.75%

Question 20: Setting Up Investment System of Equations

Question: An investment earns 5% on one amount and 6% on another. If total interest is $870, set up a system of equations to find the amounts.

Answer:

• Let x = the amount invested at 5%
• Let y = the amount invested at 6%

The system of equations would be:

• Equation 1 (Total Investment): x + y = Total Invested Amount (This value is not given in the question, so it remains a variable or placeholder)
• Equation 2 (Total Interest): 0.05x + 0.06y = $870

Question 22: Estimating Linear Cost

Question: If the cost of producing 20 units is $290, and cost increases linearly, estimate the cost for 25 units.

Answer: To estimate the cost for 25 units, you would typically use a linear cost function: Cost = Fixed Cost + (Variable Cost per Unit × Quantity).

To solve this, you would need at least two data points (cost at two different quantities) to determine the fixed and variable costs, or one point and either the fixed cost or variable cost per unit. With only one data point (20 units at $290), the specific cost for 25 units cannot be determined without additional information about the cost structure.

Question 23: Calculating Final Account Balance

Question: If your starting account balance is $1,200 and you deposit $300 and withdraw $450, what’s the final balance?

Answer: $1,200 + $300 - $450 = $1,050

Question 25: Calculating Real Income

Question: Nominal income in 2015 is $57,000. Price index = 118.75. What is real income?

Answer:

• Real Income = Nominal Income / (Price Index / 100)
• Real Income = $57,000 / (118.75 / 100) = $57,000 / 1.1875 = $48,000

Question 26: Real Income Formula

Question: State the formula for calculating real income when comparing nominal income across years using a price index.

Answer: Real Income = Nominal Income / (Price Index / 100)

Question 28: Unit Price Comparison

Question: You compare two brands:

• Brand A: $12.99 for 3L
• Brand B: $10.99 for 2.5L

Which is cheaper per liter?

Answer:

• Brand A cost per liter = $12.99 / 3L = $4.33 per liter (rounded to two decimal places)
• Brand B cost per liter = $10.99 / 2.5L = $4.396 per liter (rounded to three decimal places)

Therefore, Brand A is cheaper per liter.