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Hotra/Mathematics
Table of Contents
- Calculation
- Common notes
- Numbers
- Order of calculations
- Powers
- Units
- Variation
- Directly proportional
- Inversely proportional
- Currency
- Concepts attached with currency
- Percentage
- Per cent
- Percentage increase/decrease
- Taxation
- Value-added tax
- Income tax and other compulsory payments
- Equations
- Linear equations
- Examples of linear equations
- Quadratic equations
- Power equations
- Exponential equations
- Rates
- Concepts
- Simple rate
- Terms of payment
- Compound interest
- Margins
- Profit margin
- Critical point
- Fixing the price
- Pricing
Common notes
Although everyone thinks mathematics is universal language, this is not always the case. During the years of teaching international groups, I have noticed several ways for notations etc. What is obvious for me, may not be obvious for you and vice versa. Something is used here in Finland, something etc. in French-speaking countries. Here are some examples: [br][list][*][color=#0000ff]decimal numbers:[/color] [br][list][*]1,2 in Finland and Russia[br][/*][*]1.2 almost everywhere else[/*][/list][/*][/list][br][br][list][color=#0000ff][*]multiplication:[/*] [list][*][math]\LARGE 2 \times 3 = 2\cdot 3=2.3[/math][/*][/list][/color][/list][br][br][list][*][color=#0000ff]dividing:[/color] [list][*][math]\LARGE 2 \div 3=2/3=2:3[/math][/*][/list][/*][/list][list][/list][list][/list] [br][br]The point is that we have to find a common language to understand each other. Because you are studying in Finland, you have to also understand the other teachers, mainly from Finland, and THEY HAVE TO UNDERSTAND YOU. This means you should use Finnish notations; not always ones you have used in your home countries. [br][br]
Directly proportional
[size=150]Whenever one ratio and one of the others is known, the unknown can be solved. For two quantities with the same proportion, we say that [i][color=#0000ff]one [/color][/i][color=#0000ff]([/color][color=#0000ff][i]y[/i][i])[/i][/color][i][color=#0000ff] is proportional to the second [/color][/i][color=#0000ff]([i]x[/i])[/color] and it is written symbolically as[br] [br] [math]\huge \textcolor{blue}{y=kx,}[/math][br] [br]where [i]k[/i] is the constant of proportionality. The ratio can be expressed as a line through the origin. [br][quote][br][size=150]The more of x the more of y.[/size][/quote][br][/size][br][size=100][size=150]It can be also thought as an [u]mathematical equation[/u], where two ratios of two quantities are equal: [/size][/size][br][size=150][br] [math]\huge \textcolor{blue}{\frac{A_1}{B_1}=\frac{A_2}{B_2}}[/math][/size][br][br][size=150]If any three of these parameters are known, the missing value can be solved with cross-multiplication as[br][br] [math]\LARGE A_1B_2=B_1A_2.[/math][/size]
Example 1
[size=150]The map scale is 1:15000. If the distance of the two houses in the map is about 5 cm, what is the distance in reality? [br] [br]The ratio of the real measure and the map measure has to be the same 1:15000 all the time. That is [br][br] [math]\LARGE \frac{\text{measure in map}}{\text{reality}}=\frac{1}{15000}.[/math][br][br]We know the map measure to be 5 cm = 50 mm and measures are usually given in millimetres, so [br][br] [math]\LARGE \frac{50 \text{mm}}{x}=\frac{1}{15000}[/math][br][br] [math]\LARGE x=50 \text{mm}\cdot 15000=75000\text{ mm}=750 \text{ m}[/math][br][/size]
Example 2.
A restaurant-keeper bought 4.2 kilos of strawberries with 16.50 euros. Amount of 380 g of these strawberries were used for a cake. How much should be marked as a product cost for strawberries of the cake?[br][br]The proportion of the price and amount of strawberries must be the same. Thus,[br][br][math]\LARGE \begin{eqnarray}\frac{\text{product cost}}{380\text{ g}}&=&\frac{16.50\text{ euros}}{4200\text{ g}}\\[br]\vspace{15mm}[br]\text{product cost}&=&\frac{16.50\text{ euros}}{4200\text{ g}}\cdot 380\text{ g}=1.49 \text{ euros}[br]\end{eqnarray}[/math]
Example 3.
A recipe for 6 persons requires 500 grams of minced meat. There were 23 persons coming to the party, so how much minced meat should be bought at least?[br][br][math]\LARGE \begin{eqnarray}\frac{\text{x}}{23}&=&\frac{500\text{ grams}}{6}\\[br]\vspace{15mm}[br]x&=&\frac{500\text{ grams}}{6}\cdot 23= 1917\text{ grams}\approx 1.9 \text{ kg}[br]\end{eqnarray}[/math]
Example 4.
How much is 28 590 Japanese jeni in euros, if one euro is 122.04 JPY? Change reward is not paid attention.[br][br][math]\LARGE \begin{eqnarray}\frac{\text{x}}{1\text{ euro}}&=&\frac{28950\text{ JPY}}{122.04\text{ JPY}}\\[br]\vspace{15mm}[br]x&=&\frac{28950\text{ JPY}}{122.04\text{ JPY}}\cdot 1\text{ euro}= 237.22\text{ euros}\end{eqnarray}[/math]
Example 5.
[size=200][size=150][size=100]Linda and Ann[/size] [/size][/size]both needed some extra wine glasses at the same time. They noticed, that they can get them cheaper, if they rent a bigger amount of them and divide them. Linda needed 250 extra wine glasses and Ann 130. The total price for the rent 165 euros including all extra costs and value-added tax. If they share the total price in the same ratio as they needed glasses, how much should each of them to pay?[br][br]Let us solve first the cost for Linda. As Linda needed 250 from total of 380 glasses, we know the ratio and we can form the proportion:[br][br][math]\LARGE \frac{250}{380}=\frac{\text{cost}}{165\text{ euros}}\;\;\Leftrightarrow\;\;\text{cost}=\frac{250}{380}\cdot 165\text{ euros}=108.55\text{ euros}[/math].[br][br][size=150]Thus, Linda pays 108.55 euros and Ann has to pay 165 euros - 108.55 euros = 56.45 euros. [/size]
Example 6.
The area of an apartment and the maintanance charge of the apartment are direct variation: the bigger the apartment, the more you have to pay. As the below applet shows us, the graph expressing the direct variation is the straight line that goes through the origin. [br][br]In the graph A is the area of the apartment in m[sup]2[/sup] and B is the monthly maintanance charge in euros.
Concepts attached with currency
[b]Swift code:[/b] International three-letter code for a currency, that is used in official payments transaction. It is EUR for euro.[br][br]European Central bank (ECB) publishes the most important exchange rates outside euro area (European states using euro as their currency) every day at their homepage [url=https://www.ecb.europa.eu/stats/exchange/eurofxref/html/index.en.html][color=#0066cc]www.ecb.europa.eu[/color][/url]. Banks and companies practising currency exchange, rates their exchange rates by utilising ECB's rates. [br][br][b]Average rate:[/b] Exchange rate by ECB[br][b]Cash rate: [/b]Exchange rates for money (tourists' cash)[br][b]Commercial transactions:[/b] These are used in transactions between companies and in bank giros (bank transfers). [br][br][b]Buying rate[/b] tells at which price the bank is buying currency (from you).[br][b]Selling price[/b] tells at which price the bank sells currency (for you).[br][br]Exchange rates are informed in indirect quotation or in direct quotation. In [b]indirect quotation,[/b] it is informed, how much one euro is in other currency. In [b]direct quotation, [/b]it is informed the value of one currency in euros. [br][br]Exchange rates are different among exchange companies as their are collecting profit from their sales. Furthermore, you usually have to pay some fixed fee for an exchange. The fee is ever fixed amount of cash (e.g. 3.70 euros/exchange) or some percentage of the amount exchanged (e.g. 1.3% of exchanged cash).
Exchange rates for cash
[br]
Example 1.
How much will you get Swedish crown with 650 euros?[br][br]Now, Forex is selling you the notes and your amount is in direct variation to the exchange rate:[br][br][math]\Large\begin{eqnarray}[br] \frac{x}{650\text{ EUR}}&=&\frac{1\text{ SEK}}{0.1107 \text{ EUR}}\\[br]\vspace{10mm}[br]x&=&\frac{650}{0.1107}\text{ SEK}=5871.72 \text{ SEK}\end{eqnarray}[/math]
Example 2.
How much would a Swedish tourist get euros with 730 SEK?[br][br][math]\Large\begin{eqnarray}[br] \frac{730\text{ SEK}}{x}&=&\frac{1\text{ SEK}}{0.1032 \text{ EUR}}\\[br]\vspace{10mm}[br]x&=&730\cdot 0.1032\text{ EUR}=75.34 \text{ EUR}\end{eqnarray}[/math]
Commercial transactions
Example 3.
A company has a claim of 35000 CZK. How much is it in euros?[br][br]In this case, the bank is buying korunas from the company. Because the claim is not cash but commercial transaction, rates are looked from above table. [br][br][math]\Large\begin{eqnarray}[br] \frac{35000\text{ CZK}}{x}&=&\frac{27.547\text{ ZCK}}{1 \text{ EUR}}\\[br]\vspace{10mm}[br]x&=&\frac{35000}{27.547}\text{ EUR}=1270.56 \text{ EUR}\end{eqnarray}[/math][br]
Example 4.
A company must pay a bill of 28730 THB. How much is it in euros?[br][br]Now, the bank is selling bahts for the company, so we must use selling rates.[br][br][math]\Large\begin{eqnarray}[br] \frac{28730\text{ THB}}{x}&=&\frac{36.548\text{ THB}}{1 \text{ EUR}}\\[br]\vspace{10mm}[br]x&=&\frac{28730}{ 36.548}\text{ EUR}=786.09 \text{ EUR}\end{eqnarray}[/math]
Example 5.
The value of a currency is informed with with ECB's rate. In the below table, there is shown value of one euro in US dollars and Japanese yens.
[br][math]\Large\begin{eqnarray}[br] \frac{50\text{ USD}}{x}&=&\frac{1.0679\text{ USD}}{1 \text{ EUR}}\\[br]\vspace{10mm}[br]x&=&\frac{50}{ 1.0679}\text{ EUR}=46.82 \text{ EUR}\end{eqnarray}[/math][br][br][math]\Large\begin{eqnarray}[br] \frac{x}{50\text{ EUR}}&=&\frac{1.0679\text{ USD}}{1 \text{ EUR}}\\[br]\vspace{10mm}[br]x&=&50\cdot 1.0679\text{ EUR}=53.40 \text{ USD}\end{eqnarray}[/math]
Per cent
[size=100][size=150][color=#0000ff]Per cent[/color] means one of a hundred and it comes from latin ([i]per centum, pro centum[/i]). [br][br] [math]\huge \textcolor{blue}{1 \text{ per cent}=1\%=\frac{1}{100}=0.01.}[/math][br] [br]It is very much used in every field of life: loans, taxes, sales etc. Without undestanding its meaning, one's life can be very difficult. [br] [br]The formula for percentage is [br][br] [math]\huge \textcolor{blue}{p=\frac{b}{a}\;\;\;\text{or}\;\;\;p=\frac{100b}{a}\%,}[/math][br] [br]where [i][color=#0000ff]b[/color][/i][color=#0000ff] "=" [/color][i][color=#0000ff]p[/color][/i][color=#0000ff] but in units [/color](like kg, euro, etc.) and [i][color=#0000ff]a[/color][/i][color=#0000ff] is the basic value [/color](original value, quantity of mixture). [/size][/size]
Example 1.
There are 12500 people working in marketing and sales in an international company. The total number of workers in the company is 21630. How many per cents of workers are in marketing and sales?[br][br]In this case, [i]p [/i]is asked. As [i]b[/i] refers to same as [i]p[/i] but only in units, then [i]b[/i] = 12500. The basic value [i]a[/i] = 21630.[br][br] [math]\Large p=\frac{b}{a}=\frac{12500}{21630}=0.578=57.8\%[/math]
Example 2.
The label of a wine bottle of 750 ml informs that the percentage of alcohol is 13%. How much pure alcohol is there in the bottle?[br][br]The basic value is the volume of the bottle, so [i]a[/i] = 750 ml. The percentage of alcohol is 13%, so [i]p [/i]= 0.13. As [i]b[/i] is the amount of pure alcohol, then[br][br][math]\Large b=pa=0.13\cdot 750\text{ ml}=97.5\text{ ml}.[/math]
Example 3.
A carpet is in sale. The price of the carpet is 39 euros and it is said that price is decreased by 43%. What was the original price of this carpet?[br][br]As the price is decreased by 43%, the new price is 100% - 43% = 57% of the original price. Thus, [i]p[/i] = 57% and [i]b[/i] = 39 euros. The basic value is asked, so[br][br][math]\Large a=\frac{b}{p}=\frac{39\text{ euros}}{0.57}= 68.42\text{ euros}\approx 68\text{ euros}[/math]
Example 4.
A drink called [i]Kelkka[/i] consists of [br][br]2 cl Absolut Kurant[br]2 cl Passoã Passion Fruit[br]10 cl Orange juice.[br][br]What is the percentage in alcohol for this drink?[br][br]Absolut Kurant has 40% of alcohol and Passoã Passion Fruit has 17% of alcohol. Thus, there is pure alcohol in the drink[br][br][math]\Large b=\;0.40\cdot 2\text{ cl }+ 0.17\cdot 2\text{ cl } = 1.14 \text{ cl.}[/math][br][br]The total amount of the drink is [math]\Large a=\; 2\text{ cl } + 2\text{ cl }+ 10\text{ cl }= 14\text{ cl.}[/math][br][br]So, the percentage in alcohol is [br][br][math]\Large p=\frac{b}{a}=\frac{1.14\text{ cl}}{ 14\text{ cl}}=0.081 \approx 8\%.[/math][br]
Value-added tax
[quote]Value added tax (VAT) is a consumption tax that the seller of goods or services must add to the price. Sellers collect VAT from their customers and pay it on to the State. Liability to pay VAT concerns anyone who sells goods, services, rents out goods, or is engaged in similar operations in the conduct of business. Similarly, the primary production activity of farmers and forest owners is VAT liable. [br][br]VAT must be collected from the buyer each time a good or service is sold. Sellers add the VAT to the price they charge for goods or services. Then they pay the VAT they receive to the Tax Administration. The VAT [br]included in the price of goods and services is deductible for a VAT taxpayer who buys it for the purpose of taxable business. This requires that both the buyer and the seller are VAT taxpayers. Ultimately, the [br]consumers pay the VAT; and the final price that they pay only contains a single debiting of VAT.[br][br]VAT rates on goods and services [br] [br] [table][tr][td] 24% [br] [/td][br] [td]the general rate: most goods and services[/td][/tr][tr][td]14%[br][/td][br] [td]a reduced rate: food, animal feed, restaurant and catering services [br][/td][br][/tr][br] [tr][br] [td]10%[/td][br] [td]a reduced rate: books, pharmaceutical products, physical exercise services, film showings, entrance to cultural and entertainment events, passenger transport services, accommodation services, operations relating to TV and public broadcasting against a fee [/td][br][/tr][br][/table] [br][br][url=https://www.vero.fi/en-US/Companies_and_organisations/VAT]Finnish Tax Administration (2017)[br][/url][/quote][br]An entepreneour does not need to pay it from his/her own account but [b]VAT is added to the selling price. [/b] The entepreneour just transmits the tax from a customer to the government. In Finland, the price label already includes VAT but in some countries prices in labels are without VAT, for example in USA, and it is calculated when paid.[br][br][br]
Linear equations
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Equations in general
[color=#0000ff]Equation means equality for two expressions.[/color] An equation is [color=#0000ff]equivalent with the original equation, if[/color] [br][list][*][color=#0000ff]the same number is added[/color] for both sides of the equation,[/*][br][*]the same number is [color=#0000ff]subtracted[/color] from both sides of the equation,[/*][br][*]both sides are [color=#0000ff]multiplied or divided[/color] with the same number which is [color=#0000ff]NOT zero[/color].[/*][/list][br][br]
[size=100][color=#0000ff][size=150][b]Standard form:[/b][/size][/color][/size][br] [br] [math]\LARGE\textcolor{blue}{ax+b=0\;\;\;a,b\in \cal R,\;a\neq 0}[/math][br][br]where [i]x [/i]is the variable and [i]a[/i] and [i]b[/i] are parameters (values unknown but constant). The solution (the root of equation) is [br] [math]\LARGE\textcolor{blue}{x=-\frac{b}{a},}[/math][br] [br]where [math]\Large a \neq 0[/math] . If [i]a[/i] = 0 and [i]b[/i] = 0, then any value of satisfies the equation (identically true).[br]If [i]a[/i] = 0 and , the equation has no root (identically false).[br] [br]Solving linear equations [br][list=1][*]Remove all brackets and denominators.[/*][br][*]Transpose the equation so that terms with variable are in one side of the equation and constants are on the other side.[/*][br][*]Combine like terms for to get the equation into a form ax = b.[/*][br][*]Divide by the multiplier of the variable.[/*][br][*]Check the root by substituting it to the original equation.[/*][/list]
Concepts
[color=#0000ff][i]Capital [/i][/color](principal, pääoma) is the amount of money, for which the interest is paid for: a loan, a deposit, an investment or a mature bill.[br][br][color=#0000ff][i]Interest[/i][/color] (=korko) is either paid or received compensation of the loan money. It is given in currency. The [color=#0000ff][i]rate [/i] [/color]is the same in percentage.[br][br][color=#0000ff][i]Interest rate [/i][/color](=korkokanta) is the ratio of the interest per time. It is given in percentage and it is usually as annual interest.[br][br][color=#0000ff](Interest)[i] time[/i][/color] (=korkoaika) is the number of days between two dates for mature interest. The first date is not calculated to the interest time but the last date is. [br][br]
If interest rate is given as annual interest, then interest time must be given in years. There are three possible case for solving this:[br][br][list=1][*][b]real/360: [/b]There are 360 days in a year and the number of days is actual number of days. (ECB)[/*][*][b]real/365 (or 366): [/b]Actual number of days in a year and the number of days is actual number of days. (Typical in Finland for savings)[b][/b][/*][*][b]30/360[/b]: There are 30 days in a month and 360 in a year. The least used method but it is used for calculating residual tax. [b] [br][/b][/*][/list]There are no actual differences in the first two, as the method is paid attension when notifying the interest rate. Account and loan terms define which method is used.
Example 1.
A short-term credit was drawn January 20 and was paid at once the same year in March 15 (not a leap-year). What is the interest time in years in each cases.[br][br][table][tr][td][/td][td][b]1) real/360 [/b][/td][td][b]2) real/365 [/b][/td][td][b]3) 30/360 [/b][/td][/tr][tr][td]January[/td][td]31-20 =11[/td][td]31-20 =11[/td][td]30-20 =10[/td][/tr][tr][td]February[/td][td]28[/td][td]28[/td][td]30[/td][/tr][tr][td]March[/td][td]15[/td][td]15[/td][td]15[/td][/tr][br][tr][td]Total[/td][br][td]54[/td][br][td]54[/td][br][td]55[/td][/tr][br][tr][td]interest time [/td][br][td][math]\Large\frac{54}{360}[/math][/td][br][td][math]\Large\frac{54}{365}[/math][/td][br][td][math]\Large\frac{55}{360}[/math][/td][/tr][/table]
Profit margin
Companies' main object is profitability, which is evaluated with profit margin. Profit margin is normally used for short-term analysis (e.g. a day, a week, a month, or a year). In profit margin calculations, total costs are shared to fixed and variable costs. Profit margin can be expressed either with currency or percentages. Percentages are better for comparing different companies or different products and services inside the company.[br][br]In profit margin calculation, it is always used [b]prices without value added tax[/b]. Value added taxes just goes through the company from a customer to the government. They have no effect on company's profitability. Thus, this is valid also for costs. If a company pays some VAT when buying and gets some VAT when selling, it needs to pay only the difference for the government. [br][br]
[b]Sales proceeds = unit price [math]\cdot[/math] quantity.[/b] Sales proceeds tells, how much money flows in. [br][br][b]Variable costs[/b] changes in the same relation to the sales and production. Variable costs are, for example, material costs, cost prices of selling products, energy used etc. [br][br][b]Fixed costs[/b] do not change in the relation to the production or sales. They could be retail and /or equipment rent, fixed salaries, rates, marketing costs etc.[br][br][b]Profit margin= Sales proceed - variable costs.[/b] If profit margin is expressed in per cents (=profit margin ratio, PMR), it will tell which proportion of the unit price stays in the company after variable costs. [br][br] [math]\Large \textcolor{blue}{PMR=\frac{\text{profit margin}}{\text{sales proceeds}} \cdot 100\%}[/math][br][br][b]Result[/b] will tell, whether the company is profitable. If the result is negative, the company makes loss. If the result is expressed in per cents, the result is divided with sales proceeds:[br][br] [math]\Large \textcolor{blue}{\text{Result(\%)}=\frac{\text{result}}{\text{sales proceeds}} \cdot 100\%}[/math]
Example 1.
A group of students arranged an event. Tickets were be sold at 15 euros/ticket. The retail rent was 250 euros and salaries for fellow students 300 euros in total. They also noticed, that they used 4 euros/customer for snacks and drinks. [br] [br]Value added tax for events is 10% in Finland. Thus, the unit price for tickets used in sales proceeds is [br][br][math]\Large\begin{eqnarray}[br]x+0.1x&=&15€\\[br]1.1x&=&15€\\[br]x&=&13.64€[br]\end{eqnarray}[/math][br][br] [table] [tr][br] [td]Sold tickets[br][/td][br] [td]50[/td][br][/tr][br] [tr][br] [td]Selling price[br][/td][br] [td]15€ with VAT [/td][br] [td]13.64€ without VAT[/td][br][/tr][br] [tr][br] [td]Salaries[/td][br] [td]300[/td][br][/tr][tr][td]Snacks and drinks[/td][br] [td]4€/customer[/td][/tr][tr][br] [td]Rent[/td][br] [td]250€[/td][br][/tr][br][/table][br][br][table][tr][td][b]Sales proceeds[/b] [/td][td][math]\Large 50\cdot 13.64€=[/math][/td][td][b]682€[/b][/td][/tr][tr][td]-Variable costs[/td][td][math]\Large 50\cdot 4€=[/math][/td][td]200€[/td][/tr][tr][td][b]=Profit margin[/b][/td][td][/td][td][b]482€[/b][/td][td][b]71%[/b][/td][/tr][tr][td]-Fixed costs[/td][td][/td][td]550€[/td][/tr][tr][td][b]=Result[/b][/td][td][/td][td][b]-68€ [/b][/td][td][b]-10%[/b][/td][/tr][/table][br]The profit margin in this example is quite good but fixed costs are too high for a small event.
Example 2.
Profit margin of a product is 30%. What is the new profit margin in percentages, if the price is reduced by 20%.[br][br]Variable costs are 70%. The price reduce affects only to sales proceeds but not in variable costs. If the original price is [i]p[/i], then the new price is 80% of [i]p[/i] = 0.8[i]p[/i]. Now, we can write a direct variation to solve the new variable costs in percentage:[br][br] [math]\Large\begin{eqnarray}[br]\frac{\text{New sales proceeds}}{\text{Variable costs}}&=&\frac{100\%}{x}&=&\frac{p€}{0.7p€}\\[br]\frac{0.8p}{0.7p}&=&\frac{100\%}{x}\\[br]x&=&\frac{100\cdot 0.7p}{0.8p}&=&87.5\%[br]\end{eqnarray}[/math][br][br]Thus, the new sales proceeds is 100% - 87.5%=12.5%