# Practice Problems

##### Large Number Multiplication Problems

### Overview

- Candidates in Case Interviews frequently need to multiply large numbers when calculating Revenue, or in Marketing Sizing and Estimation problems. The numbers involved are usually large — practice your large number multination skills with the problems below, and register for the FastMath Ace the Case Online Course to learn mental calculation methods for these types of problems.

- 4
*Thousand*× 50 - 8
*Million*× 4*Thousand* - 20
*Million*× 125 - 60
*Thousand*× 3*Thousand* - 400
*Thousand*× 7*Thousand* - 24
*Million*× 5*Thousand* - 160
*Million*× 250 - 20
*Million*× 7*Thousand* - 40
*Million*× 350 - 84
*Thousand*× 500

- 280
*Thousand*× 75 - 440
*Thousand*× 150 - 360
*Million*× 25 - 240
*Million*× 750 - 25
*Million*× 60*Thousand*× 15% - 40
*Million*× 6*Thousand*× 25% - 80
*Million*× 45*Thousand*× 40% - 200
*Million*× 15*Thousand*× 12% - 2.5
*Thousand*× 12*Thousand*× 15 - 320
*Thousand*× 50 × 6

##### Percentage Multiplication Problems

**Calculate 50% of:**

- $20
- $600
- $5,000
- $15
*Thousand* - 45
*Million* - $34
*Million* - 63
*Thousand* - $3,250
- $126
*Million* - 1.6
*Million*

**Calculate 10% of:**

- 30
- $900
- $43
*Million* - $12.5
*Million* - 8.4
*Million* - $4.5
*Billion* - 720
*Thousand* - $5.75
*Million* - $29.7
*Million* - 2.4
*Million*

**Calculate 25% of:**

- $200
*Million* - $24
*Million* - $600
*Million* - $480
*Million* - $520
*Million* - $1.6
*Million* - $260
*Thousand* - $410
*Million* - 1.4
*Billion* - $560
*Million*

- 60
*Million* - $800
*Million* - $70
*Million* - $45
*Million* - 8.4
*Million*

**Calculate 5% of:**

- $50
*Billion* - 2.5
*Million* - $64
*Million* - $42.5
*Billion* - 7.5
*Million*

**Calculate 1% of:**

**Calculate:**

- 20% of 50
*Million* - 30% of 400
*Million* - 60% of 30
*Million* - 15% of 80
*Million* - 75% of £4.8
*Billion* - 90% of $600
*Million* - 70% of $450
*Million* - 20% of $3.2
*Billion* - 95% of $8
*Billion* - 80% of $350
*Million*

- 18% of 50
*Million* - 85% of $400
*Million* - 2% of $340
*Million* - 6% of $420
*Million* - 45% of $240
*Million* - 12% of $35
*Million* - 23% of $12
*Billion* - 35% of $160
*Million* - 18% of $45
*Million* - 38% of $150
*Million*

##### Profitability and Breakeven Analysis Problems

- A company sells smartphone cases. They sell each case for $20, and each case costs $10 to manufacture. Their factory costs $200 K per year to operate.
- How many units would they need to sell to have annual profit of $1 M?
- How many units would they need to sell to have annual profit of $2 M?
- What would their profit be if they sold 500 K units per year?

- A company manufactures microscopes for $1,200. Each microscope costs $700 to manufacture. The company spends $800 pear year on factory rent and maintenance, and $200 K per year in labor costs.
- How many units does the company need to sell annually to break even?
- How many units would they need to sell to have annual profit of $10 M?
- How many units would they need to sell to have annual profit of $20 M?
- What would their profit be if they sold 60 K units per year?

- A company sells blenders for $85 per unit. Each blender costs $45 in materials, $5 in energy costs, and $10 in incremental labor costs. It costs the firm $1.5 M in rent and equipment maintenance per year to run the factory. Administrative salaries are $300 K per year, and other operational costs are $200 K per year.
- How many units does the company need to sell annually to break even?
- How many units would they need to sell to have annual profit of $6 M?
- How many units would they need to sell to have annual profit of $15 M?
- What would their profit be if they sold 800 K units per year?
- What would their profit be if they sold 1.2 M units per year?

- A smoothie store sells smoothies for an average price of $6.50. Each smooth costs $1.25 in ingredients, and $0.25 for packaging. The store spends $7,500 per month on rent, $2,500 per month on utilities and $10,000 per month on labor.
- How many smoothies does the store need to sell monthly to break even?
- How many smoothies do they need to sell to have $25 K in monthly profit?
- How many smoothes does the store need to sell to have in monthly profit?
- If they sell 12 K smoothies per month, what will their monthly profit be?

- A company manufactures wooden tables, which they sell for $200 per table. Each table costs $90 in materials and $35 in labor to manufacture. The company spends $125 K per month for rent and utilities on their factory, and $25 in other overhead per month.
- How many units will then need to sell monthly to break even?
- How many units will they need to sell to have monthly profits of $600 K?
- What will their monthly profit be if they sell 14 K units per month?

- A company produces and sells canned fruit. They sell each can of fruit for $4.50, and each can costs $1.50 in raw materials and $0.50 in packaging costs. They spend $150 K per month on factory rent, $15 K per month on utilities, and $35 K per month for labor.
- How many units do they need to sell monthly to break even?
- How many units do they need to sell to have monthly profit of $500 K?
- How many units do they need to sell to have monthly profit of $2 M?
- What would their annual profit be if they sold 500 K units per month?

- A company manufactures and sells designer jewelry. Each piece of jewelry sells for an average of $350, and costs $165 materials and $35 in labor. Every year, they spend $250 K in rent and maintenance on their factory, $40 K in other operational expenses, and $10 K on advertising.
- How many units does the company need to sell annually to break even?
- How many units would they need to sell to have annual profit of $12 M?
- How many units would they need to sell to have annual profit of $18 M?
- What would their profit be if they sold 240 K units per year?
- What would their profit be if they sold 300 K units per year?

- A company manufacturers computers which they sell for $1,500 per computer. For each computer sold, the company incurs incremental costs of $900 for parts, $100 in labor, and $50 in transportation costs. Per year, they spend $2 M on factory rent and maintenance, and $250 K in utility costs.
- How many units do they need to sell annually to break even?
- How many units do they need to sell annually have a profit of $9 M?
- What would their annual profit be if they sold 65 K units per year?

- A company manufactures and sells motorized scooters. Each scooter sells for $4,750 on average, and costs $2,750 in parts and material, and $500 in labor to manufacture. They spend $45 M on factory rent, utilities, and maintenance per year, $7 M on other administrative overhead, and $8 M on advertising per year.
- How many units does the company need to sell annually to break even?
- How many units would they need to sell to have annual profit of $300 M?
- How many units would they need to sell to have annual profit of $1.2 B?
- What would their profit be if they sold 6 M units per year?
- What would their profit be if they sold 2.8 M units per year?

- A company manufacturers injectable vaccines. Each vaccine sells for $530 and costs $180 to manufacture. Their production facility costs $1.15 M to operate annually. They spend $400 K per year on sales and marketing expenses, and $200 K per year on regulatory compliance.
- How many units does the company need to sell annually to break even?
- How many units would they need to sell to have annual profit of $21 M?
- How many units would they need to sell to have annual profit of $49 M?
- What would their profit be if they sold 400 K units per year?

##### Market Sizing and Estimation Problems

- How many school buses are there in the United States?
- How many weddings occur in the United States annually?
- How many gas stations are in the US?
- How many disposable baby diapers are consumed annually and what is the corresponding Market Size?
- In the United States?
- In the United Kingdom
- In Germany?

- How many boxer shorts (and boxer-briefs) are sold annually in the US?
- How many units of alpine ski and snowboards were sold last year:
- In Norway?
- In Germany?
- In The United States?

- What’s the Market Size for cholesterol drugs in the United States?
- What is the Market Size for Burrito’s in the United States?
- What’s the Market Size for adult blue jeans in the United States?
- How much food would a Tyrannosaurus Rex consume each day?

##### Inverse Proportion Problems

- A company sells a product for $30 per unit and sells 5 M units per year. If they reduce the Price by 20% what percentage increase in Quantity do they need in order to achieve the same Revenue. What is the numerical increase in units sold required to break-even on Revenue?
- A company sells a product for $120 per unit sells 35 M units per year. If they decrease Price by $20, what percentage increase in Quantity do they need in order to achieve the same Revenue? What is the corresponding numerical increase in units sold?
- A company cuts the price of a product by 10%. What change in Quantity do they need in order to achieve the same Revenue?
- A company increases Price by 25%, how much of their existing Quantity sold can they afford to lose and keep their Revenue constant? That is, what reduction in Quantity will result in Revenue remaining the same? Initial tests indicate Quantity will drop by 22.5%, will their Revenue increase or decrease?
- A company increases Quantity produced by 20% which increase supply and hence reduces Price. What percentage decrease in Price results in neutral Revenue?
- A company increases Quantity produced by 60%. What percentage decrease in Price results in neutral Revenue? Initial research suggests their Price will be reduced by 40%, will they increase or decrease revenue?
- A company sells 480 M units of a product per year for $20. If they decrease their production by 40 M units, what Price change would keep their revenue constant?
- A company sells a product for $1,800 per unit sells 48 M units per year. If they decrease Price by $600, what percentage increase in Quantity do they need in order to achieve the same Revenue? What is the corresponding numerical increase in units sold?
- A manufacturing company has a problem with a particular machine which reduces overall worker productivity by 25%.
- The company can hire temporary workers who will have the same productivity as their existing employees. How many temporary workers will they need to hire in order to keep production constant?
- The company can also have existing workers increase the number of hours worked. How many additional hours will workers need to work in order to keep production constant?

- A company sells a product for $140 per unit sells 125 M units per year. If they increase Price by $35, what change in Quantity would lead to Revenue remaining constant? What production volume does this correspond to? If the Price increase causes Quantity to by 30 M units annually, will their Revenue increase or decrease?
- A company cuts Price of a product by 40%. What change in Quantity do they need in order to achieve the same Revenue?
- A company increases Price of a product by 40%. How much of their existing Quantity sold can they afford to lose and keep their Revenue constant? If Quantity drops by 30%, will they increase or decrease Revenue?
- A company decreases the Price of a Product from $600 to $450, and their units sold increase annually from 140 Million units to 175 Million units. Will this increase or decrease revenue?
- If the company instead increases Price from $600 to $750 and decreases Quantity from 140 Million units to 120 Million units, will this increase or decrease Revenue? What change in Quantity results in Revenue remaining constant?
- American Airlines is thinking of changing the price of airplane tickets from New York to LA. The current cost is $450. Existing average number of seats sold is 76% of airplane capacity — this is called the Load Factor (LF).
- If American Airlines increases ticket price from $450 to $500, what percentage of seats would need to be sold under the new ticket price to keep Revenue constant? That is, what is the new Load Factor required to keep Revenue constant?
- If American Airlines decreases from $450 to $400 USD, what percentage of seats would need to be sold under the new ticket price to keep Revenue constant? That is, what is the new Load Factor required to keep Revenue constant?

- An ultra-marathon runner decreases the time it takes them to run a 60 mile race from 10 hours to 8 hours. How much faster are the running than before?