Quantitative Reasoning Chapter 5 MTH105

Chapter 5 covers problem solving which is the process of finding solutions to difficult, challenging, or complex issues or questions. It involves using critical thinking, logic, creativity, and analytical skills to identify and implement effective solutions to problems. Problem-solving is a fundamental skill in various aspects of life, including:

1. Everyday Life: We encounter and solve problems in our daily routines, such as fixing a broken appliance, planning a trip, or managing our time efficiently.

2. Education: Problem-solving is a key component of education, helping students develop critical thinking skills and apply what they've learned to practical situations.

3. Workplace: Problem-solving is essential in the workplace, where employees often face challenges related to their tasks, projects, and collaboration with colleagues.

4. Business: Entrepreneurs and business leaders rely on problem-solving to address issues like market competition, product development, and financial management.

5. Science and Research: Scientists and researchers use problem-solving to design experiments, analyze data, and make discoveries.

6. Engineering and Technology: Engineers and technologists solve complex problems when designing new technologies, infrastructure, and systems.

Problem-solving typically involves the following steps:

1. Identify the Problem: Clearly define the problem or challenge you're facing. What needs to be solved or improved?

2. Gather Information: Collect relevant data and information related to the problem. Research and gather facts.

3. Generate Solutions: Brainstorm potential solutions or strategies to address the problem. Consider various options.

4. Evaluate Solutions: Assess the pros and cons of each solution. Analyze their feasibility and potential outcomes.

5. Choose a Solution: Select the best solution based on your evaluation. Sometimes, a combination of solutions may be necessary.

6. Implement the Solution: Put your chosen solution into action. Execute the plan.

7. Monitor and Adjust: Continuously monitor the results of your solution. If necessary, make adjustments or improvements.

Problem-solving often involves critical thinking, creativity, adaptability, and the ability to work well with others in collaborative efforts. It's a skill that can be developed and refined over time, and it's valuable in various aspects of personal and professional life.

Unit 5 Exercises - Problem Solving:

Employees in 2012 paid 4.2% of their gross wages towards social security (FICA tax), while employers paid another 6.2%. How much will someone earning $61,000 a year pay towards social security out of their gross wages?

$Correct

Give your answer accurate to at least the nearest dollar.

To calculate how much someone earning $61,000 a year will pay towards Social Security (FICA tax) out of their gross wages, you can follow these steps:

1. Find the employee's share of the FICA tax:

   • The employee pays 4.2% of their gross wages.
   • Gross wages = $61,000

   Employee's share = 4.2% of $61,000 = (0.042) * $61,000 = $2,562

2. So, someone earning $61,000 a year will pay $2,562 towards Social Security out of their gross wages.

3. 
   

The population of a town increased from 3350 in 2008 to 5200 in 2009. Find the absolute and relative (percent) increase.

Absolute increase: Correct

Relative increase: Correct %

Give answers accurate to at least 1 decimal place.

To find the absolute increase, you can subtract the initial population (in 2008) from the final population (in 2009):

Absolute increase = Final population - Initial population Absolute increase = 5200 - 3350 Absolute increase = 1850

So, the absolute increase in population is 1850.

To find the relative (percent) increase, you can use the following formula:

Relative increase (%) = (Absolute increase / Initial population) * 100

Plug in the values:

Relative increase (%) = (1850 / 3350) * 100 Relative increase (%) ≈ 55.22%

The relative increase in population is approximately 55.22%.

A company's sales in Seattle were $400,000 in 2012, while their sales in Portland were $285,000 for the same year. Complete the following statements:

a. Seattle's sales were Correct% larger than Portland's.
b. Portland sales were Correct% smaller than Seattle's.
c. Portland sales were Correct% of Seattle's.

Give answers accurate to at least one decimal place.

a. Seattle's sales were 40% larger than Portland's.

To calculate this percentage increase, you can use the following formula:

$\text{Percentage Increase}=\left(\frac{\text{Seattle Sales}-\text{Portland Sales}}{\text{Portland Sales}}\right)\times 100$

Plugging in the values:

$\text{Percentage Increase}=\left(\frac{\mathrm{\$}400,000-\mathrm{\$}285,000}{\mathrm{\$}285,000}\right)\times 100=\left(\frac{\mathrm{\$}115,000}{\mathrm{\$}285,000}\right)\times 100\approx 40.35\mathrm{\%}$

So, Seattle's sales were approximately 40.35% larger than Portland's. Rounded to 40.4

b. Portland sales were 28.75% smaller than Seattle's.

To calculate this percentage decrease, you can use the following formula:

$\text{Percentage Decrease}=\left(\frac{\text{Seattle Sales}-\text{Portland Sales}}{\text{Seattle Sales}}\right)\times 100$

Plugging in the values:

$\text{Percentage Decrease}=\left(\frac{\mathrm{\$}400,000-\mathrm{\$}285,000}{\mathrm{\$}400,000}\right)\times 100=\left(\frac{\mathrm{\$}115,000}{\mathrm{\$}400,000}\right)\times 100=28.75\mathrm{\%}$

So, Portland sales were 28.75% smaller than Seattle's. Rounded to 28.8

c. Portland sales were 71.25% of Seattle's.

To calculate this percentage, you can use the following formula:

$\text{Percentage}=\left(\frac{\text{Portland Sales}}{\text{Seattle Sales}}\right)\times 100$

Plugging in the values:

$\text{Percentage}=\left(\frac{\mathrm{\$}285,000}{\mathrm{\$}400,000}\right)\times 100=0.7125\times 100=71.25\mathrm{\%}$

So, Portland sales were 71.25% of Seattle's. Rounded to 71.3

A store has clearance items that have been marked down by 25%. They are having a sale, advertising an additional 40% off clearance items. What percent of the original price do you end up paying?

Correct%

Give your answer accurate to at least one decimal place.

To find out what percent of the original price you end up paying after the initial 25% markdown and an additional 40% off clearance items, you can calculate it step by step:

1. After the initial 25% markdown, you're paying 100% - 25% = 75% of the original price.
2. Then, with an additional 40% off, you're paying 100% - 40% = 60% of the price after the first markdown.

So, you end up paying 60% of the price after the first markdown. In terms of the original price, this is:

$60\mathrm{\%}\text{ of }75\mathrm{\%}=(0.60)\times (0.75)=0.45$

To express this as a percentage, you can multiply by 100:

$0.45\times 100=45\mathrm{\%}$

So, you end up paying 45% of the original price after the initial 25% markdown and an additional 40% off clearance items.

A car is driving at 40 kilometers per hour. How far, in meters, does it travel in 2 seconds?

Correct meters

Give your answer to the nearest meter.

To find out how far the car travels in 2 seconds while driving at 40 kilometers per hour, you can use the following steps:

1. Convert the speed from kilometers per hour to meters per second.
2. Multiply the speed in meters per second by the time in seconds to find the distance.

Let's break it down:

1. Convert speed from kilometers per hour (km/h) to meters per second (m/s):

   1 km = 1000 meters (since there are 1000 meters in a kilometer) 1 hour = 3600 seconds (since there are 3600 seconds in an hour)

   So, to convert from km/h to m/s, you can use the conversion factor:

   $1\text{ km/h}=\frac{1000\text{ m}}{3600\text{ s}}=\frac{5}{18}\text{ m/s}$

   Therefore, the speed of 40 km/h is equivalent to:

   $40\text{ km/h}=40\times \frac{5}{18}\text{ m/s}=\frac{200}{18}\text{ m/s}\approx 11.11\text{ m/s}$

2. Multiply the speed in meters per second by the time in seconds to find the distance:

   Distance (in meters) = Speed (in meters per second) × Time (in seconds)

   $��������=11.11\text{ m/s}\times 2\text{ s}=22.22\text{ meters}$

So, the car travels approximately 22.22 meters in 2 seconds at a speed of 40 kilometers per hour.

You want to put a 6 inch thick layer of topsoil for a new 29 ft by 15 ft garden. The dirt store sells by the cubic yards. How many cubic yards will you need to order? The store only sells in increments of 1/4 cubic yards.

Correct cubic yards

1. Thickness of the topsoil layer: 6 inches = 6/12 yards = 0.5 yards (since 1 yard = 36 inches).
2. Garden length: 29 feet = 29/3 yards (since 1 yard = 3 feet).
3. Garden width: 15 feet = 15/3 yards.

Now, calculate the volume of topsoil required:

Volume = (29/3 yards) × (15/3 yards) × (0.5 yards) = 8.25 cubic yards.

You read online that a 15 ft by 20 ft brick patio would cost about $2,275 to have professionally installed. Estimate the cost of having a 25 by 26 ft brick patio installed.

$Correct

Round your answer to the nearest dollar.

To estimate the cost of having a 25 ft by 26 ft brick patio installed, you can use the ratio of the areas between the two patios and apply it to the cost.

First, calculate the area of the original 15 ft by 20 ft brick patio:

Area = Length × Width = 15 ft × 20 ft = 300 square feet.

Next, calculate the area of the new 25 ft by 26 ft brick patio:

Area = Length × Width = 25 ft × 26 ft = 650 square feet.

Now, find the ratio of the areas:

Area ratio = New patio area / Original patio area = 650 sq. ft. / 300 sq. ft. = 2.167.

Now, apply this ratio to the cost of the original patio:

Estimated cost = Original cost × Area ratio = $2,275 × 2.167 ≈ 4,929

So, the estimated cost of having a 25 ft by 26 ft brick patio installed would be approximately $4,929.

Out of 280 racers who started the marathon, 251 completed the race, 23 gave up, and 6 were disqualified. What percentage did not complete the marathon?

Correct %

Give your answer accurate to at least 1 decimal place.

To find the percentage of racers who did not complete the marathon, you can sum the number of racers who gave up and were disqualified, and then divide that by the total number of racers who started the marathon. Finally, multiply by 100 to express the answer as a percentage.

Number who did not complete = Number who gave up + Number who were disqualified Number who did not complete = 23 + 6 = 29

Now, calculate the percentage:

Percentage who did not complete = (Number who did not complete / Total number who started) × 100 Percentage who did not complete = (29 / 280) × 100 ≈ 10.36%

So, approximately 10.36% of the racers did not complete the marathon. Rounded 10.4

A company wants to decrease their energy use by 13%. If their electric bill is currently $2,600 a month, what will their bill be if they are successful? Give your answer accurate to at least the nearest dollar.

$Correct

If the company's electric bill for next month is $2314, should that be considered a success?

• No, that is not a success
• Yes, that is a success

To calculate the company's new electric bill if they successfully decrease their energy use by 13%, follow these steps:

1. Calculate the 13% reduction amount:

   Reduction = 13% of the current bill Reduction = 0.13 * $2,600 = $338

2. Subtract the reduction from the current bill to find the new bill:

   New bill = Current bill - Reduction New bill = $2,600 - $338 = $2,262

So, if the company is successful in decreasing their energy use by 13%, their new monthly electric bill will be $2,262.

Now, let's consider the bill for the next month, which is $2,314. To determine if this should be considered a success, you can compare it to the new bill after the 13% reduction:

$2,314 (next month's bill) > $2,262 (new bill after 13% reduction)

Since the next month's bill is higher than the new bill after the reduction, it should not be considered a success. The company's goal was to reduce their bill by 13%, but the new bill is still higher than that target.

An article reports "attendance dropped 7% this year, to 3300." What was the attendance before the drop?

Correct people

Round to the nearest whole person.

To find the attendance before the 7% drop, you can use the following steps:

1. First, express the drop as a percentage of the original attendance.

   7% can be represented as 0.07 in decimal form.

2. Let $�$ represent the attendance before the drop.

3. Set up an equation to represent the drop:

   $�-0.07�=3300$

4. Simplify the equation:

   $0.93�=3300$

5. Solve for $�$ by dividing both sides by 0.93:

   $�=\frac{3300}{0.93}$

Now, calculate $�$:

$�\approx 3548.39$

So, the attendance before the 7% drop was approximately 3548.39. Since attendance typically consists of whole numbers, you can round it to the nearest whole number:

The attendance before the drop was approximately 3,548 people.

A company's sales in Seattle were $350,000 in 2012, while their sales in Portland were $250,000 for the same year. Complete the following statements:

a. Seattle's sales were Correct% larger than Portland's.
b. Portland sales were Correct% smaller than Seattle's.
c. Portland sales were Correct% of Seattle's.

Give answers accurate to at least one decimal place.

Let's complete the statements based on the given sales figures:

a. Seattle's sales were 40% larger than Portland's.

To calculate this percentage increase, you can use the following formula:

$\text{Percentage Increase}=\left(\frac{\text{Seattle Sales}-\text{Portland Sales}}{\text{Portland Sales}}\right)\times 100$

Plugging in the values:

$\text{Percentage Increase}=\left(\frac{\mathrm{\$}350,000-\mathrm{\$}250,000}{\mathrm{\$}250,000}\right)\times 100=\left(\frac{\mathrm{\$}100,000}{\mathrm{\$}250,000}\right)\times 100=40\mathrm{\%}$

So, Seattle's sales were 40% larger than Portland's.

b. Portland sales were 20% smaller than Seattle's.

To calculate this percentage decrease, you can use the following formula:

$\text{Percentage Decrease}=\left(\frac{\text{Seattle Sales}-\text{Portland Sales}}{\text{Seattle Sales}}\right)\times 100$

Plugging in the values:

$\text{Percentage Decrease}=\left(\frac{\mathrm{\$}350,000-\mathrm{\$}250,000}{\mathrm{\$}350,000}\right)\times 100=\left(\frac{\mathrm{\$}100,000}{\mathrm{\$}350,000}\right)\times 100=28.57\mathrm{\%}$

So, Portland sales were approximately 28.57% smaller than Seattle's.

c. Portland sales were 71.43% of Seattle's.

To calculate this percentage, you can use the following formula:

$\text{Percentage}=\left(\frac{\text{Portland Sales}}{\text{Seattle Sales}}\right)\times 100$

Plugging in the values:

$\text{Percentage}=\left(\frac{\mathrm{\$}250,000}{\mathrm{\$}350,000}\right)\times 100=\left(\frac{\mathrm{\$}5}{\mathrm{\$}7}\right)\times 100=71.43\mathrm{\%}$

So, Portland sales were 71.43% of Seattle's.

It takes a hose 4 minutes to fill a rectangular aquarium 9 inches long, 11 inches wide, and 14 inches tall. How long will it take the same hose to fill an aquarium measuring 22 inches by 24 inches by 30 inches?

Correct minutes

Round your answer to the nearest minute

To find how long it will take the hose to fill the larger aquarium, you can use the following steps:

Calculate the volume of the first aquarium (in cubic inches):

Volume = Length × Width × Height

Volume = 9 inches × 11 inches × 14 inches = 1,386 cubic inches

Calculate the rate at which the hose fills the first aquarium:

Rate = Volume / Time

Rate = 1,386 cubic inches / 4 minutes = 346.5 cubic inches per minute

Calculate the volume of the larger aquarium (in cubic inches):

Volume = Length × Width × Height

Volume = 22 inches × 24 inches × 30 inches = 15,840 cubic inches

Calculate the time it will take to fill the larger aquarium using the rate calculated in step 2:

Time = Volume / Rate

Time = 15,840 cubic inches / 346.5 cubic inches per minute ≈ 45.7 minutes

So, it will take approximately 45.7 minutes for the same hose to fill the larger aquarium measuring 22 inches by 24 inches by 30 inches.

A 6 inch personal pizza has 610 calories, with 240 of those from fat. A 12 inch pizza is cut into 8 slices. Estimate the number of calories in one slice of a 12 inch pizza. Hint... must use Area = Pi*r^2 proportions to solve this problem.. DO NOT SOLVE BY PIECE.

Correct calories

Round your answer to the nearest calorie.

A circle's area goes up with the square of its radius (A = πr²). Since the 12-inch pizza has twice the radius of the six-inch pizza it has 2² = 4 times the area. In addition, since the 12-inch pizza is then cut into 8 slices, each slice has 4/8 = 1/2 times the area of the 6-inch pizza, so each slice has approximately 1/2*610 = 305 calories.

When Ibuprofen is given for fever to children 6 months of age up to 2 years, the usual dose is 5 milligrams (mg) per kilogram (kg) of body weight when the fever is under 102.5 degrees Fahrenheit. How much medicine would be usual dose for a 18 month old weighing 19 pounds?

Correct milligrams

Round your answer to the nearest milligram.

To calculate the usual dose of Ibuprofen for a child weighing 19 pounds, you'll need to convert the weight from pounds to kilograms since the dose is given in milligrams per kilogram (mg/kg).

1 pound is approximately 0.453592 kilograms.

So, for a 19-pound child:

Weight in kilograms = 19 pounds * 0.453592 kg/pound ≈ 8.63 kg

Now, you can calculate the usual dose:

Usual dose = Weight (in kg) * Dose (in mg/kg) Usual dose = 8.63 kg * 5 mg/kg ≈ 43.15 mg

Therefore, the usual dose of Ibuprofen for an 18-month-old child weighing 19 pounds would be approximately 43.15 milligrams. Please note that it's important to consult with a healthcare professional before giving any medication to a child, as dosages may vary based on specific guidelines and the child's medical condition.

A long year-end status report for work is 155 pages long. You need to print 17 copies for a meeting next week. How much is the paper going to cost for those reports? Paper is sold in reams (500 pages) for $3.40 each.

$Correct

Give your answer to the nearest cent, only for the paper you use (partial reams are OK)

Total pages needed: 17 copies * 155 pages/copy = 2,635 pages.

Calculate the number of reams required:

Reams = Total pages / Pages per ream

Reams = 2,635 pages / 500 pages/ream ≈ 5.27 reams (rounded up to the nearest whole ream).

So, you need 6 reams of paper to print 17 copies of the 155-page report.

Calculate the cost of paper:

Cost per ream = $3.40

Total cost = Number of reams * Cost per ream

Total cost = 5.27 reams * $3.40/ream ≈ 17.92

You need to buy some chicken for dinner tonight. You found an ad showing that the store across town has it on sale for $3.29 a pound, which is cheaper than your usual neighborhood store, which sells it for $3.39 a pound. Is it worth the extra drive?

• How much chicken will you be buying? 6 pounds
• How much far are the two stores? My neighborhood store is 2.5 miles away, and takes about 8 minutes. The store across town is 8.6 miles away, and takes about 23 minutes.
• How kind of mileage does your car get? It averages about 22 miles per gallon in the city.
• How many gallons does your car hold? About 15 gallons
• How much is gas? About $3.80/gallon right now.

• Going to the close store is cheaper
• Going to the further store is cheaper

Correct

The cheaper option saves you $1.51

To determine whether it's worth the extra drive to the store across town for cheaper chicken, let's calculate the costs for both options:

Option 1: Buying chicken from your usual neighborhood store

• Cost per pound: $3.39
• Amount of chicken needed: 6 pounds
• Total cost = Cost per pound × Amount of chicken = $3.39 × 6 = $20.34

Option 2: Buying chicken from the store across town

• Cost per pound: $3.29
• Amount of chicken needed: 6 pounds
• Total cost = Cost per pound × Amount of chicken = $3.29 × 6 = $19.74

Now, let's calculate the cost of driving to each store:

For your neighborhood store (2.5 miles away):

• Distance: 2.5 miles (one way)
• Gas mileage: 22 miles per gallon
• Gas cost (one way) = Distance / Gas mileage = 2.5 miles / 22 mpg ≈ 0.1136 gallons
• Round trip gas cost = 2 * 0.1136 gallons ≈ 0.2272 gallons
• Gas price per gallon: $3.80
• Total gas cost (round trip) = Round trip gas cost × Gas price per gallon ≈ 0.2272 gallons × $3.80/gallon ≈ $0.86

For the store across town (8.6 miles away):

• Distance: 8.6 miles (one way)
• Gas mileage: 22 miles per gallon
• Gas cost (one way) = Distance / Gas mileage = 8.6 miles / 22 mpg ≈ 0.3909 gallons
• Round trip gas cost = 2 * 0.3909 gallons ≈ 0.7818 gallons
• Gas price per gallon: $3.80
• Total gas cost (round trip) = Round trip gas cost × Gas price per gallon ≈ 0.7818 gallons × $3.80/gallon ≈ $2.97

Now, let's compare the total costs:

Option 1 (neighborhood store):

• Chicken cost: $20.34
• Gas cost: $0.86
• Total cost: $20.34 + $0.86 = $21.20

Option 2 (store across town):

• Chicken cost: $19.74
• Gas cost: $2.97
• Total cost: $19.74 + $2.97 = $22.71

Based on these calculations, going to your usual neighborhood store is the cheaper option, and it saves you approximately $1.51 compared to going to the store across town. Therefore, it's not worth the extra drive for cheaper chicken in this scenario.

A friend has a 78% average before the final exam for a course. That score includes everything but the final, which counts for 20% of the course grade.

What is the best course grade your friend can earn? Correct%

What is the minimum score your friend would need on the final to earn a 75% for the course? Correct%

Give answers accurate to at least one decimal place.

To determine the best course grade your friend can earn and the minimum score needed on the final to earn a 75% for the course, we can set up the following equations:

1. Best Course Grade: Let $�$ be the best course grade. The final exam counts for 20%, so the weighted average for the course after the final would be: $0.78\times 80\mathrm{\%}+�\times 20\mathrm{\%}=�$

To get the best possible grade, we want to maximize $�$, so we set $�=100\mathrm{\%}$ (since 100% is the highest possible grade):

$�=100\mathrm{\%}$

So, the best course grade your friend can earn is 100%.

2. Minimum Score on the Final: Let $�$ be the minimum score your friend would need on the final exam to earn a 75% for the course. We can set up the equation as follows:

$0.78\times 80\mathrm{\%}+�\times 20\mathrm{\%}=75\mathrm{\%}$

Now, solve for $�$:

$0.78\times 0.80+0.20�=0.75$ $0.624+0.20�=0.75$

Subtract 0.624 from both sides:

$0.20�=0.75-0.624$ $0.20�=0.126$

Now, divide by 0.20:

$�=\frac{0.126}{0.20}=0.63$

To earn a 75% for the course, your friend would need to score a minimum of 63% on the final exam.

Must show calculations/formula for credit. Any answer given with no calculations shown will result in no credit for that answer.

 

 

1.     A newspaper headline claims: "Apple spends far less on Research & Development than any of its rivals." Further into the story, the reporter writes that Apple spends a paltry 2% of revenues compared with 14% at Google and Microsoft.

 

a.    What is the mistaken assumption here?

That both companies have the same revenue but the newspaper only compares the percentages of the two companies not the actual data difference.

b.    Why is that assumption incorrect?

     Both of these companies will have a difference when it comes to revenue   numbers so how much they spend on research can’t be compared based on percentages.

c.    Is there a situation where the headline is still correct?  Only if the revenue of both of the companies was the same then the headline would be correct.

2.     A shop owner raises the price of a $120 pair of shoes by 30%. After a few weeks, because of falling sales, the owner reduces the price of the shoes by 30%. A customer then says that the shoes are back at the original price.

 

a.    What is the mistaken assumption here? 

b.    Why is that assumption incorrect?

The final price is now 36 which isn’t the same as the 120

c.    What do the shoes actually cost now?

             

 

d.    By what percent should the shoes be decreased in order to have the price back at $120? Round to the nearest 10^th percentage. (for example if your decimal answer is .058267 your answer would be 5.8267% round to nearest 10^th percent answer is 5.8%) show calculation

a. The mistaken assumption here is that raising the price of a product by a certain percentage and then reducing it by the same percentage will bring the price back to its original level.

b. The assumption is incorrect because percentage increases and decreases are applied to the current price, not the original price. When you increase the price of the shoes by 30%, you're adding 30% of the current price to it. When you later decrease the price by 30%, you're subtracting 30% of the current price from it, which results in a lower final price compared to the original price.

c. To find the actual cost of the shoes now, we need to calculate the new price after both the increase and decrease:

1. Increase the price by 30%: New price = $120 + (30% of $120) New price = $120 + ($36) New price = $156

2. Decrease the price by 30%: New price = $156 - (30% of $156) New price = $156 - ($46.80) New price = $109.20

So, the shoes actually cost $109.20 now.

d. To find out by what percentage the shoes should be decreased to have the price back at $120, you can set up the following equation:

$��������=�������������\times (1-\mathrm{\%}��������\mathrm{/}100)$

Substitute the known values:

$120=156\times (1-\mathrm{\%}��������\mathrm{/}100)$

Now, solve for the percentage decrease:

$\frac{120}{156}=1-\mathrm{\%}��������\mathrm{/}100$

$\frac{120}{156}-1=-\frac{\mathrm{\%}��������}{100}$

$-0.2308=-\frac{\mathrm{\%}��������}{100}$

To isolate % decrease, multiply both sides by -100:

$\mathrm{\%}��������=0.2308\times 100$

$\mathrm{\%}��������=23.08\mathrm{\%}$

So, the shoes should be decreased by approximately 23.08% to have the price back at $120.

           

3. The annual number of burglaries in a town rose by 40% in 2012 and fell by 10%

in 2013. Hence the total number of burglaries increased by 30% over the two-

year period.

a. What is the mistaken assumption here?

b. Why is that assumption incorrect?

c. By what percent has the number of burglaries actually changed in the

two-year period? show calculation

d. By what percent would the crime have to decrease in the second year in

order for the change over the two-year period to actually be a 30%

increase? Round to nearest 10th percent (ex-decimal .05873 is 5.873% to

one decimal is 5.9%) show calculation

a. The mistaken assumption here is that a 40% increase in one year followed by a 10% decrease in the next year results in a net increase of 30% over the two-year period.

b. The assumption is incorrect because percentage changes are based on the previous value. When you have a 40% increase followed by a 10% decrease, you are not simply adding and subtracting percentages from the starting point. Percentage changes compound, and you need to apply them sequentially. In this case, a 40% increase is applied to the original number, and then a 10% decrease is applied to the increased number. This sequential application of percentages leads to a different result.

c. To calculate the actual change in the number of burglaries over the two-year period:

1. Calculate the increase in 2012: Increase = 40% of the previous year's value Increase = 40% of the original value Increase = 0.4 * original value

2. Calculate the decrease in 2013: Decrease = 10% of the increased value from 2012 Decrease = 10% of (original value + 0.4 * original value) = 0.1 * (original value + 0.4 * original value)

3. Calculate the actual change over the two-year period: Actual change = Increase - Decrease Actual change = 0.4 * original value - 0.1 * (original value + 0.4 * original value)

Now, let's calculate:

Actual change = 0.4 * original value - 0.1 * (original value + 0.4 * original value) Actual change = 0.4 * original value - 0.1 * (1.4 * original value) Actual change = 0.4 * original value - 0.14 * original value Actual change = 0.26 * original value

So, the actual change in the number of burglaries over the two-year period is 26% (not 30%).

d. To find out by what percent the crime would have to decrease in the second year for the change over the two-year period to be a 30% increase, we can set up the following equation:

$30\mathrm{\%}=40\mathrm{\%}-�\mathrm{\%}\text{(where x is the decrease percentage in the second year)}$

Now, solve for x:

$�\mathrm{\%}=40\mathrm{\%}-30\mathrm{\%}=10\mathrm{\%}$

So, the crime would have to decrease by 10% in the second year for the change over the two-year period to be a 30% increase.

4. A store is currently offering a 50% discount on all items purchased. Your cashier

is trying to convince you to open a store credit card and says to you, "In addition

to the 50% discount you are receiving for purchasing these items on sale today,

you will get an additional 25% off for opening a credit card account. That means

you are getting 75% off!"

a. What is the mistaken assumption here?

b. Why is that assumption incorrect?

c. If you did truly have 75% discount, explain what should happen when

you go to the counter to buy $500 worth of items? show calculation

d. If you got your 50% discount and opened the card for an additional 25%,

what is the actual % discount you would receive? show calculation

e. Is it better to apply the 50% discount first or the 25% discount first? show

calculation

a. The mistaken assumption here is that the additional 25% off for opening a store credit card account can be added directly to the 50% discount, resulting in a total discount of 75%.

b. The assumption is incorrect because percentage discounts do not work by simply adding them together. When you apply a percentage discount, it is calculated based on the original price, and subsequent discounts are applied to the reduced price. In this case, the additional 25% discount should be applied to the price after the initial 50% discount, not to the original price.

c. If you truly had a 75% discount, here's what should happen when you go to the counter to buy $500 worth of items:

1. Calculate the initial 50% discount: Discount = 50% of $500 Discount = 0.50 * $500 = $250

2. Calculate the final price after the initial discount: Final price = Original price - Discount Final price = $500 - $250 = $250

So, with a 75% discount, the final price for $500 worth of items should be $250.

d. To find the actual percentage discount you would receive when you get your 50% discount and open the card for an additional 25%, you can calculate it as follows:

1. Calculate the price after the 50% discount: Price after 50% discount = Original price - (50% of Original price) Price after 50% discount = $500 - (0.50 * $500) = $250

2. Calculate the additional 25% discount: Additional discount = 25% of Price after 50% discount Additional discount = 0.25 * $250 = $62.50

3. Calculate the final price after both discounts: Final price = Price after 50% discount - Additional discount Final price = $250 - $62.50 = $187.50

Now, let's calculate the percentage discount:

Percentage discount = [(Original price - Final price) / Original price] * 100% Percentage discount = [($500 - $187.50) / $500] * 100% Percentage discount ≈ 62.50%

So, the actual percentage discount you would receive is approximately 62.50%, not 75%.

e. It is better to apply the 50% discount first and then the 25% discount because percentage discounts are calculated based on the current price. Applying the 50% discount first reduces the initial price significantly, and then the 25% discount is applied to the reduced price, resulting in a lower final price compared to applying the discounts in the opposite order.