Comparing Quantities

overview

** Comparing Numbers ** : When comparing two or more numbers, statements like one is greater than or less than another do not provide the relative magnitutes or amounts. The formal ways to specify the comparison are ratios, proportions, and percentage.

comparing numbers

The distance to a town $P$ is $120$ km and to another town $Q$ is $240$ km.

It is noted that the two numbers can be compared and say that the town $Q$ is farther than town $P$.

Town $Q$ is calculated to be farther by *comparing the numbers* $120$ and $240$.

In comparing two numbers $p$ and $q$, only one of the following is true.

• $p$ equals $q$

• $p$ is greater than $q$

• $p$ is lesser than $q$

In this case, $p=120$ is lesser than $q=240$.
*This is called trichotomy property of numbers. *

shortcoming

Comparing two numbers $p=120$ and $q=240$, it is given than $q$ is greater.

Consider the numbers $r=1212$ and $s=120000$. It is evident that all the numbers $q$, $r$, and $s$ are greater than $p$.

When saying that the numbers $q$, $r$, and $s$ are greater than $p$, some information is lost. That is, magnitude of the numbers, whether the numbers are almost equal, or far greater is not available in the statement.

Apart from saying $q$ is greater, the following helps to understand the numbers better.

"$q$ is double of $p$". If $p$ is multiplied by $2$, we get $q$.
$120\times 2=240$

relative magnitudes

Comparing numbers $p=120$ with $q=240$, $r=122$ and $s=120000$.

$q$ is $2$ times $p$

$r$ is $1.1$ times $p$ (*You will learn about decimal $1.1$ in some time.*)

$s$ is $1000$ times $p$.

Now the relative magnitudes of $q$, $r$, $s$ are easily understood.

There are formal ways to specify the comparison. Those are ratios, proportions, and percentage.

two forms of numbers

*Numbers represent quantities.*

• The number of pens in the box is $2$

• Length of a rope is $1\frac{1}{2}$ meters

• $\frac{1}{4}$th of the apple is remaining
*Numbers are also used to compare quantities.*

• The number of pens in the red box is $2$ times the number of pens in the blue box

• Length of rope is $1\frac{1}{2}$ times the height of the table.

• The number of apples he had is $\frac{1}{4}$th of the number she had.

context

When a number is specified, a context is provided and the meaning of the number is defined in the context.

For example, Consider:

Length of a rope is $1\frac{1}{2}$ meter. The context in this specifies the measure of the quantity.

Length of the rope is $1\frac{1}{2}$ times the height of the table. The context in this specifies the comparison of one quantity to another.

While learning ratios, proportions, and percentage, the context becomes very important. The specifics of the context is explained in the due course of this lesson.

summary

** Comparing Numbers ** : When comparing two or more numbers, statements like one is greater than or less than another do not provide the relative magnitutes or amounts. The formal ways to specify the comparison are ratios, proportions, and percentage.

Outline

The outline of material to learn "commercial arithmetics" is as follows.

Note: * Click here for the detailed ouline of commercial arthmetics.*

• **Ratio, Proportion, Percentage**

→ __Comparing Quantities__

→ __Introduction to Ratio__

→ __Ration & Fraction Differences__

→ Proportions__P__

→ __Percentages__

→ __Conversion to percentage__

• **Unitary Method**

→ __Introduction to Unitary Method__

→ __Direct Variation__

→ __Inverse Variation__

→ __DIV Pair__

• **Simple & Compound Interest**

→ __Story of Interest__

→ __Simple Interest__

→ __Compound Interest__

• **Rate•Span=Aggregate**

→ __Understanding Rate-Span__

→ __Speed • Time=Distance__

→ __Work-rate • time = Work-amount__

→ __Fill-rate • time = Filled-amount__

• **Profit-Loss-Discount-Tax**

→ __Profit-Loss__

→ __Discount__

→ __Tax__

→ __Formulas__