maths > commercial-arithmetics

Understanding Rate•Span=Aggregate

what you'll learn...

overview

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

speed $×$$\times$ time $=$$=$ distance
$20$$20$m/sec $×3$$\times 3$ sec $=60$$= 60$ m

DIV pair

Consider
At price $2$$2$ coins per banana,
the price of $3$$3$ bananas
is overall cost of $6$$6$ coins.

If the number of bananas is increased to $4$$4$, the overall cost increases to $8$$8$ coins.

Number of bananas and overall cost are in direct variation.

Consider
At price $2$$2$ coins per banana,
the price of $3$$3$ bananas
is overall cost of $6$$6$ coins.

If the price is increased to $3$$3$ coins per banana, the overall cost increases to $9$$9$ coins.

The price of bananas and overall cost are in direct variation.

Consider
At price $2$$2$ coin per banana,
the price of $3$$3$ bananas
is overall cost of $6$$6$ coins.

If the price is increased to $3$$3$ coins per banana, the number of bananas one can buy reduces to $2$$2$.

The price of bananas and number of bananas are in inverse variation.

Note that the direct and inverse variations come as a pair in the same equation.

Multiplicand $×$$\times$ Multiplier $=$$=$ Product

"Multiplicand" and "Product" are in direct variation.

"Multiplier" and "Product" are in direct variation.

"Multiplicand" and "Multiplier" are in inverse variation.

rate-formula as DIV pair

The direct and inverse variations pair.

Multiplicand $×$$\times$ Multiplier $=$$=$ Product

A particular form of rate-formula is common and is repeated in different forms.
rate $×$$\times$ span $=$$=$ aggregate.

Various forms of this equation is explained in the subsequent pages.

Note: "aggregate" means sum over multiple items. The multiplication is understood as repeated addition.

price•number=cost

Consider
At price $2$$2$ coin per banana,
the price of $3$$3$ bananas
is the overall cost of $6$$6$ coins.

The price can be given as rate of money per unit.

price $×$$\times$ number $=$$=$ cost

•  "price" is the rate of money per unit

•  "number" is the span

•  "cost" is the aggregate of money

»  Price and cost are related by direct variation.

»  Number and cost are related by direct variation.

»  Price and number are related by inverse variation.

speed•time=distance

Consider
At speed $2$$2$ meter per second,
in time $3$$3$ seconds,
the distance traveled is $6$$6$ meter.

The speed can be given as rate of distance per unit time.

speed $×$$\times$ time $=$$=$ distance

•  "speed" is the rate of distance per unit time

•  "time" is the duration span

•  "distance" is the aggregate of speed-distance-time over time

»  Speed and distance are related by direct variation.

»  Time and distance are related by direct variation.

»  Speed and time are related by inverse variation

work-rate•time=work-done

Consider that a person is building a wall.
At work-rate $2$$2$ meter per day,
in time $3$$3$ days,
the overall length of wall built is $6$$6$ meter.

The work-rate can be given as rate of work done per unit time.

work rate $×$$\times$ time $⇒$$\Rightarrow$ work done

•  "work rate" is the rate of work per unit time

•  "time" is the duration span

•  "work done" is the aggregate of work over the time

»  Work rate and work done are related by direct variation.

»  Time and work done are related by direct variation.

»  Work rate and time are related by inverse variation

In most time and work problems, the "work done" is constant and so the work rate and time are in inverse variation.

fill-rate•time=filled-amount

Consider a pipe filling a cistern or a tank.
At fill-rate $2$$2$ liter per second,
in time $3$$3$ seconds
the amount filled is $6$$6$ liters.

The fill rate can be given as rate of volume per unit time.

fill rate $×$$\times$ time $⇒$$\Rightarrow$ filled amount

•  "fill rate" is the rate of volume per unit time

•  "time" is the duration span

•  "filled amount" is the aggregate of volume over time

»  Fill rate and filled amount are related by direct variation.

»  Time and filled amount are related by direct variation.

»  Fill rate and time are related by inverse variation

summarizing all

The direct and inverse variations pair.

Multiplicand $×$$\times$ Multiplier $=$$=$ Product

A rate-formula as direct and inverse variation pair.
rate $×$$\times$ span $=$$=$ aggregate.

Various forms of this equation are:
price $×$$\times$ number $=$$=$ cost
speed $×$$\times$ time $=$$=$ distance
work rate $×$$\times$ time $⇒$$\Rightarrow$ work done
fill rate $×$$\times$ time $⇒$$\Rightarrow$ filled amount

summary

Rate•Span=Aggregate:
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$$2$ coins per item $×$$\times$ $3$$3$ items = $6$$6$ coins in total

speed $×$$\times$ time $=$$=$ distance
$20$$20$m/sec $×3$$\times 3$ sec $=60$$= 60$ m

Outline