Commercial Arithmetics

Welcome to the topics in commercial arithmetics.

• comparing 2 numbers as one is greater than other does not provide the information on the *relative magnitude of the numbers*. For example, 11 and 1000 are greater than 10, but 11 is close to 10 and 1000 is far greater than 10. This relative magnitude is specified with ratio, proportion, and percentage.

• *Direct and Inverse variations come in a pair*. Learn the revolutionary insights into how a simple multiplication p*q=r leads to direct variation and inverse variation. p and r are in direct variation and p and q are in inverse variation.

• Simple and Compound Interest are simplified with couple of similar looking formulas. With this *novel explanation*, students are relieved of memorizing 10+ formulas.

• A special case of Direct and Inverse Variation pair is *rate * span = aggregate*. Speed-time-distance, time-work, and pipes-cisterns are explained in this unique lesson.

• Profit-loss, discount, and tax are simplified with one standardized formula for all. With this *extremely simple explanation*, students are relieved of memorizing 20+ formulas.

Comparing Quantities

In numerical arithmetics comparing two numbers is done by specifying one number as greater than, equal-to, or smaller than the other. That does not provide the relative magnitutes or amounts. To have a better comparison, the relative magnitude of the numbers are specified.

Numbers represent quantities. eg: length of a rope is $2$m.

Numbers are also used to Compare quantities: eg: Length of a rope is $2$ times of length of the pole.

Introduction to Ratio

In this page, specifying ratio of two quantities is explained. For example, the number $24$ is double of $12$. These two quantities are in the ratio $2$ to $1$. The symbols $:$ is introduced to specifying a ratio, eg $2:1$.

Ratio & Fraction Differences

Ratio and Fraction are two methods to specifying relative magnitude of quantities. Eg: $12$ and $24$ are in
$1:2$ ratio

OR $12$ is $\frac{1}{2}$ of $24$.

Understanding the differences in these two representation is explained.

Proportions

Two ratios are said to be in proportion, if the ratios are equivalent. For example $2:4$ and $3:6$ are equivalent. Such equivalent ratios are formally represented as a proportion. The representation is $2:4::3:6$.

Introduction to Percentages

A ratio is represented with two numbers, eg: $3:4$. Comparing two different ratios involve some arithmetics. eg: $4:5$ and $3:4$, which one is larger in ratio?

To simplify such comparison, the second term is standardized to a value, say $100$, then the given ratios are $80:100$ and $75:100$. It is easier to compare these ratios.

Such standard representation is simplified by dropping the known number $100$. and the simplified form is "percentage". eg: $80\%$ and $75\%$.

Conversion of Ratio to Percentage

In this page, conversion of ratio to percentage and fraction to percentage are explained. The most important part is to understand the context in which a ratio or fraction is given and handle the conversion accordingly.

Unitary Method : Introduction

Multiplication is

$\text{multiplicand}\phantom{\rule{1ex}{0ex}}\times \phantom{\rule{1ex}{0ex}}\text{multiplier}\phantom{\rule{1ex}{0ex}}$ $=\text{product}$

It is common to have multiplier as number of items. eg: Price of $10$ pens is $200$ coins.

When the multiplier is number of items, the multiplicand and product can be found for $1$ item. This helps in simplifying solution to finding product for any number of items.

eg: Price of $1$ pen is $20$ coins. And with that it is easy to find price of $5$ pens.

Introduction Direct Variation

Multiplication is

$\text{multiplicand}\phantom{\rule{1ex}{0ex}}\times \phantom{\rule{1ex}{0ex}}\text{multiplier}\phantom{\rule{1ex}{0ex}}$ $=\text{product}$

It is explained that

• When multiplier is not changing, the multiplicand and product are in direct variation.

• When multiplicand is not changing, the multiplier and product are in direct variation.

Introduction Inverse Variation

Multiplication is

$\text{multiplicand}\phantom{\rule{1ex}{0ex}}\times \phantom{\rule{1ex}{0ex}}\text{multiplier}\phantom{\rule{1ex}{0ex}}$ $=\text{product}$

It is explained that

• When product is not changing, the multiplicand and the multiplier are in inverse variation.

Direct and Inverse Variation Pair

The objective of this lesson is to show that direct and inverse variations are two sides of the same problem. A number of examples are provided to understand this fact.
*number of days $\times$ rate (coins per day) $=$ overall earnings*
*number of books $\times$ weight of a book $=$ weight of the parcel*

*length of cloth $\times$ rate $=$ overall cost*

*number of machines $\times$ number of hours $\Rightarrow$ number of machine-hours *

* number of pipes $\times$ number of hours $\Rightarrow$ amount filled *

*number of hens $\times$ number of days $\Rightarrow$ amount of food grain*

Story of "Interest"

In this page, the following terminology is introduced

loan,

investment,

deposit,

lender,

borrower,

principal,

interest,

interest rate,

loan duration, and

loan maturity value.

The story connecting these are made very easy to remember.

Simple Interest

Simple interest is made easy with the following.

Maturity Value $A$ = Principal $P+$ Interest $PRT$

$=P(1+R+R+R...T$ times)

Note: R is the (interest rate in percentage)/100.

The interest is calculated on the principal for each of the time period in $T$. That is, the interest $PR$ is added $T$ times.

Once the above is understood, the equation is easily derived.

Compound Interest

Compound interest is made easy with the following.

maturity value in $n$ time periods $A=P{(1+R)}^{n}$

$=P(1+R)(1+R)(1+R)...n$ times

Note: R is the (interest rate in percentage)/100.

The interest is calculated on the principal for each of the time period in $n$. That is, the principal for the second time period is the maturity value of first time period, and so on for subsequent time periods.

Once the above is understood, the equation is easily derived.

Understanding Rate•Span=Aggregate

DIV pair is given by

Multiplicand $\times$ Multiplier $=$ product

A special case of DIV pair is

rate $\times$ span $=$ aggregate.

Where,

span is the count of items or time duration and

rate is the value per unit span.

eg: price per item $\times$ number of items $=$ total cost

$2$ coins per item $\times$ $3$ items = $6$ coins in total

speed $\times$ time $=$ distance

$20$m/sec $\times 3$ sec $=60$ m

In this page, this is explained in detail.

Speed, Time, Distance

An example to the rate $\times$ span $=$ aggregate, is

speed $\times$ time $=$ distance.

This is explained in detail.

Time and Work

An example to the rate $\times$ span $=$ aggregate, is

work-rate $\times$ time $=$ work-done.

This is explained in detail.

Pipes and Cistern

An example to the rate $\times$ span $=$ aggregate, is

fill-rate $\times$ time $=$ amount-filled.

This is explained in detail for pipes and cisterns.

Introduction to Profit and Loss

This page introduces the terms cost-price, sale-price, overhead-expenses, profit, and loss. It explains that the shopkeeper invested cost-price and so profit is calculated as a percent of the amount invested.

Profit percent $=100\times$ (sale price - cost price) $/$ cost price.

Introduction to Discount

This page introduces the terms marked-price and discount. It explains that the buyer sees the marked price and has a discount on that, and so discount is calculated as a percent of the marked-price.

discount percent $=100\times$ (marked price - sale price) $/$ marked price.

Introduction to Tax

This page introduces the terms bill-price, and tax. It explains that the seller collects the tax on behalf of government on the money transacted which is the sale-price and so the tax is calculated as a percent of the sale-price.

tax percent $=100\times$ (bill price - sale price) $/$ sale price.

Formulas for profit-loss-discount

Students need not memorize 20+ formulas anymore for the topics profit-loss, discount, and tax.

$\text{Profit\%}=100\times$ $(\text{SalePrice}-\text{costPrice})$$/\text{costPrice}$

$\text{Discount\%}=100\times$ $(\text{MarkedPrice}-\text{SalePrice})$$/\text{MarkedPrice}$

$\text{Tax\%}=100\times$ $(\text{BilledPrice}-\text{SalePrice})$$/\text{SalePrice}$

All these three formulas are very similar and has a simple explanation to the terms in numerator and denominator.