Numbers English Contest #18

Using the English and data from Contest #17, what is the percent profit margin change based on production?

Calculate this margin for initial and final conditions.

Solution to Contest #17

The first step is to calculate the profit at the initial conditions, based on 200 units of sales.

Total costs = Fixed + variable = $234,200 + $137 x 200 = $261,600

Profit = Sales - Total Costs = $1490 x 200 - 261,600 = $36,400

Profit per Unit (Margin) = $36,400 / 200 = $182 per unit.


Using the same equations, we now calculate the margin with the changes:

Total costs = Fixed + variable = $(234,200+12,500) + $151 x 200 = $276,900

Profit = Sales - Total Costs = $1490 x 200 - 276,900 = $21,100

Profit per Unit (Margin) = $21,100 / 200 = $105.50 per unit.


Now we calculate the % change in margin for the increased expenses:

% Change = (Final Margin - Initial Margin) / Initial Margin x 100% = (105.50-182.00)/182.00 x 100% = -42% 


These fairly small decreases in costs (5.3% in fixed and 10.2% in variable) reduced margin by 42%. This concept is called leveraging.

Numbers English Contest #17

West Electronics is a specialty instrumentation manufacturer for water treatment plants. They have developed a loyal following in North America for a line of their sensors that sells for $1490. 

Fixed expenses are $234,200 per month. Variable expenses are $137 per unit.

If fixed costs rise by $12,500 and variable costs jump to $151, what is the change in the per unit margin (based on monthly sales of 200 units) expressed in percentage points? The selling price stays the same.

Please note that this short passage of English text requires at least six sets of calculations to get the right answer. Can you translate this English to frame the calculations properly? 

Solution to Contest #16

How much will it cost to run this test advertising campaign?

Each viewing of the ad will be seen by 5,784 (240,000 x 0.0241) people.

To get at least half million impression, it will need to be shown 87 times (500,000 / 5,784 and rounded up)

Total cost is $27,405 (87 x $315).

If the manufacturer's profit is $825 per car sold, how many cars must be sold just to pay for this advertising campaign?   

The manufacturer needs to sell at least  34 cars ($27,405 / $825 and rounded up) more than its usual sales to pay for this advertising campaign.

 

Numbers English Contest #16

A car manufacturer is testing out an advertising campaign by using a local TV station. It has identified a particular evening TV show it feels will reach its target audience. The car manufacturer wants to run a half million impressions in the next month to see how this campaign affects sales in this area. If the campaign is successful, it will expand this campaign nationally.

The area serves 240,000 people. The ratings for this TV show are 2.41% of the total population. TV commercials for that show run for $315 for a 30-second spot. How much will it cost to run this test advertising campaign? If the manufacturer's profit is $825 per car sold, how many cars must be sold just to pay for this advertising campaign?   

Solution to Contest #15

First, we take a look at what the questions is asking for.

Which campaign is most effective for Dave, based on a cost per 1000 impressions?

Note that the question did not ask for cost per click.

Next we calculate the cost per 1000 impressions for Campaign #1.

$31.00 / 40,134 x 1000 = $0.772 per thousand impressions.

Next, we do some calculations for Campaign #2. We must think a little to use the information relating to "clicks" appropriately.

Cost = $31.00 x 2.1 = $63.00

Clicks = 158

Click Through Rate = 0.92% = 0.0092

Impressions = 158/0.0092=17,174

Cost per 1000 impressions = $63.00 / 17,174 x 1000 = $3.67 per thousand impressions.

Next, we do some calculations for Campaign #3. 

Cost = $31.00 - $12.25 = $18.75

Clicks = 158 * (1-0.115) = 140

Click Through Rate = 0.92% + 0.54% = 1.46% = 0.0146

Impressions = 140/0.0146 = 9,589

Cost per 1000 impressions = $18.75 / 9,589 x 1000 = $1.96 per thousand impressions. 

Campaign #1 is the most effective campaign, based on cost per thousand impressions. 

Contest #15

Dave is evaluating the effectiveness of his Google ad campaigns for his business English program. Although he is paying on a cost per click basis, he is looking to see which campaign is giving him the most impressions for the least money.

Last month, Campaign #1 cost $31.00. It had 105 clicks and 40,134 impressions.

Campaign #2 cost two point one times as #1. It had 158 clicks and had a click-through rate of zero point nine two percent.

Campaign #3 cost $12.25 less than campaign #1. It had 11.5% fewer clicks than Campaign #1, but had a higher click-through rate than Campaign #2 by zero point five four percentage points. 

Which campaign is most effective for Dave, based on a cost per 1000 impressions?

Solution to Contest #14

• Current holding in $US = $100 + Є100 x $0.75543/Є=$175.54.
• New exchange rate = 0.75543 x (1— 0.026) = 0.73579.
• Future holdings in $US = $100 + Є100 x 0.73579 = $173.58.
• % Change = (173.58 — 175.54) / 175.54 x 100% = —1.1%.

His holding will have dropped by 1.1% by next week if this rate change occurs.    

Contest #14

Jack has US$100 and E100. Today, one US$ buys 0.75543 Euros. If this rate is expected to drop by 2.6% next week, what will be the percent change in his holdings in terms of US$?

Solution to Contest #13

This numbers English contest has lots of numbers English to translate into calculation. It also requires some logic to set up the calculation.

The question seems quite tricky:

What is the expected loss in profit from March to April?

Some of you may be asking: "How can we have a loss in profits?" After all, losses and profits are opposites of each other; we should only have one or the other, right?

To explain this, let's consider a company that has a profit of $2,000 in Month 1 and a profit of $1,500 in Month 2. There is a $500 loss in profits from Month 1 to Month 2. Now do you understand how words are used in this way?

We need to start with the end of March production figures to build our figures for March and April.

At the end of March, an oil well was producing 50.5 m3 of oil per day and 72.1 m3 of water per day. Each month, the well's oil production decreases by 1.855% and water production increases by 2.011%.

To get the figures for the end of April, we simply make these calculations:

Oil rate = 50.5 m3 x (1 - 0.01855) = 49.56 m3 per day.

Water rate = 72.1 m3 x (1 + 0.02011) = 73.55 m3 per day.

Average oil rate = (50.5 + 49.56) /2 = 50.03 m3 per day

Average water rate = (72.1 + 73.55) / 2 = 72.82 m3 per day

Total oil production = 50.03 m3 / day x 30 days = 1,500.9 m3
Total water production = 72.82 m3 / day x 30 days = 2,184.6 m3

The figures for March are going to be more difficult to calculate because the 50.5 and 72.1 are at the end of the month. In English, it is important to base the % calculation on the starting figures, which, in this case, are for the beginning of March, which is what we are trying to solve. So the calculations work like this:

Oil rate = 50.5 / (1 - 0.01855) = 51.45 m3 per day.
Water Rate = 72.1 / (1 + 0.0211) = 70.61 m3 per day.

Average oil rate = (51.45 + 50.5) / 2 = 50.98 m3 per day.
Average water rate = (70.61 + 72.1) / 2 = 71.36 m3 per day.

Monthly oil production = 50.98 x 31 days = 1580.4 m3.
Monthly water production = 71.36 x 31 days = 2,212.2 m3.

The next step is calculating the profits for March and April:

It costs $140,250 to operate this well each month.
[It also costs] $51.90 to process one cubic meter of oil and $15.50 to dispose one m3 of water.

For March the costs are: 140,250 + 51.90 x 1580.4 + 15.50 x 2212.2 = $256,563.
For April the costs are: 140,250 + 51.90 x 1500.9 + 15.50 x 2184.6 = $252,008.

Oil price is expected to be $575.00 per m3.

Supposedly, the petroleum company already knew the price of oil in March, but this figure was not provided. The "expected" price is for April. Because this information is somewhat vague, we can only use the best information we have. We will assume that the oil price for both months is $575.00 per m3.

March revenues = 1580.4 x 575.00 = $908,730
April revenues = 1500.9 x 575.00 = $863,018.

What is the expected loss in profit from March to April?

March profit = $908,730 - $256,563 = $652,157
April profit = $863,018 - $252,008 = $611,010

So the loss in profits (from March to April) is $652,157 - $611,010 = $41.147

If you were watching your calculations closely, much of this loss in profits can be attributed to March having one extra day.

Contest #13

At the end of March, an oil well was producing 50.5 m3 of oil per day and 72.1 m3 of water per day. Each month, the well's oil production decreases by 1.855% and water production increases by 2.011%. It costs $140,250 to operate this well each month, which including operator salary and amortization of equipment. Additional costs are $51.90 to process one cubic meter of oil and $15.50 to dispose one m3 of water. Oil price is expected to be $575.00 per m3. What is the expected loss in profit from March to April?

Solution to Contest #12

Bigby Co. & Figetz Inc. are manufacturing competitors. They each sell their widgets for $45.00 a piece. Bigby's fixed costs are $101,900 a month. Figetz's fixed costs are 21.1% lower than Bigby's. Per widget, Figetz has a variable cost of 25.75% of the sale price. The cost is 3.5 percentage points more than Bigby's variable costs.

Bigby's fixed costs = $101,900 per month.
Figetz's fixed costs = $101,900 x (1 - 0.211) = $80,399 per month.
Figetz's variable cost = $45.00 x 25.75/100 = $11.59 per widget.
Bigby's variable cost = $45.00 x ( 25.75 - 3.50)/100 = $10.01 per widget

Last month, Bigby produced and sold 5,123 widgets. Figetz produced and sold 145 more widgets than Bigby.

Bigby's profit = 5,123 x ($45.00 - $10.01) - $101,900 = $77,354
Figetz's profit = (5,123 + 145) x ($45.00 - $11.59) - $80,399 = $95,605

Who made more profit and what was the % difference from the manufacturer who made the lower profit?

Figetz earned 23.6% more profit than Bigby: ($95,605 - $77,354) / $77,354 x 100%

Contest #12

Bigby Co. & Figetz Inc. are manufacturing competitors. They each sell their widgets for $45.00 a piece. Bigby's fixed costs are $101,900 a month. Figetz's fixed costs are 21.1% lower than Bigby's. Per widget, Figetz has a variable cost of 25.75% of the sale price. The cost is 3.5 percentage points more than Bigby's variable costs. Last month, Bigby produced and sold 5,123 widgets. Figetz produced and sold 145 more widgets than Bigby.


Who made more profit and what was the % difference from the manufacturer who made the lower profit?

Solution to Contest #11

Adam is an accountant. His client wants him to use this depreciation schedule for a certain asset:

In the first year, the asset is depreciated 20% from its original price. In subsequent years, the asset is depreciated by 8% of its book value.

What is the book value of a $109,000 asset after its fourth year?

The asset is bought for $109,000. In its first year, the depreciation is $109,000 x 0.20 = $21,800. The book value of asset is $109,000 - $21,800 = $87,200.

In the second year, the depreciation is $87,200 x 0.08 = $6,976. The book value of asset is $87,200 - $6,976 = $80,224.

In the third year, the depreciation is $80,224 x 0.08 = $6,418. The book value of asset is $80,224 - $6,418 = $73,806.

In the fourth year, the depreciation is $73,806 x 0.08 = $5,904. The book value of asset is $73,806 - $5,904 = $67,566.

So the correct answer is $67,902.

Contest #11

Adam is an accountant. His client wants him to use this depreciation schedule for a certain asset:

In the first year, the asset is depreciated 20% from its original price. In subsequent years, the asset is depreciated by 8% of its book value.

What is the book value of a $109,000 asset after its fourth year?

Solution to Contest #10

Bill is a sales representative for a farm equipment dealership. He gets paid $3,000 a month plus one half of a percent of tractor sales over $150,000 a month. Last month, Bill sold eight tractors for a total of $702,000. How much did the dealership pay Bill for the month?

Bill's base salary is $3,000 per month, and he earns this amount regardless of how much he sells. On the first $150,000 of sales, Bill earns no commission, but commission starts accumulating after this amount. All this can be translated into this equation:

Bill's monthly pay = $3,000 + ($702,000 - $150,000) x 0.5 / 100 = $5,760.

Contest #10

Bill is a sales representative for a farm equipment dealership. He gets paid $3,000 a month plus one half of a percent of tractor sales over $150,000 a month. Last month, Bill sold eight tractors for a total of $702,000. How much did the dealership pay Bill for the month?

Solution to Contest #9

This is a two-step problem as the price of the dress is discounted twice.

Sara is an assistant manager in a clothing store. Her boss told her it is time to unload last year's fashions. He told her to reduce prices of all clothes by 25%. One red dress normally sells for $299.00.

The first discount is calculated as:

Discounted price = $299.00 x (1 - 0.25) = $224.25


It did not sell. So the boss told Sara to reduce the discounted price by 25%. What is the sale price of this dress?

The key word is "discounted." This tells us the reference for the second discount is based on the discounted price, not the original price.

Second discounted price = $224.25 x (1 - 0.25) = $168.18.

If the second discount had been stated as "So the boss told Sara to reduce the price by 25%," it would have been unclear whether the discount was to be applied to the original price (which would have given us $149.50 as the correct answer) or the discounted price. Even native speakers would be divided as to how to correctly translate this English phrase into math. Sometimes it is best to ask questions when then the "numbers English" seems a little too complicated.

Contest #9

Sara is an assistant manager in a clothing store. Her boss told her it is time to unload last year's fashions. He told her to reduce prices of all clothes by 25%.

One red dress normally sells for $299.00. It did not sell. So the boss told Sara to reduce the discounted price by 25%. What is the sale price of this dress?

Show how the English is framing the calculations to get the right answer.

Solution to Contest #8

Jack has two bank accounts that total $1,041.50.

x = first bank account, y = second bank account

x + y = 1,041.50

The first bank account has 35.1% less money than the second bank account.

x = y * (1 - 0.351) = y * 0.649

Solving:

y * 0.649 + y = 1,041.50

1.649 * y = 1,041.50

y = 631.50

So x = 1,041.50 - 631.60 = 409.90

What is the difference between the first and second bank account?

631.60 - 409.90 = 221.70

The difference between the two bank accounts is $221.70. Did you understand how the English formulated these calculations?

If the question was rephrased "The second bank account had 35.1% more money than the first," would the answer be different?