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Thought-Process to Discover Knowledge

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Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

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mathsCommercial ArithmeticsRate and Span

### Understanding Rateā¢Span=Aggregate

DIV pair is given by

Multiplicand xx Multiplier = product

A special case of DIV pair is
rate xx span = aggregate. Where,
span is the count of items or time duration and
rate is the value per unit span.

eg: price per item xx number of items = total cost
2 coins per item xx 3 items = 6 coins in total

speed xx time = distance
20m/sec xx 3 sec = 60 m

click on the content to continue..

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

If the number of bananas is increased to 4, what is the overall cost?

• 8 coins
• 8 coins
• 6 coins

The answer is "8 coins". Number of bananas and overall cost are in direct variation.

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

If the price is increased to 3 coins per banana, what is the overall cost?

• 8 coins
• 9 coins
• 9 coins

The answer is "9 coins". The price of bananas and overall cost are in direct variation.

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

If the price is increased to 3 coins per banana, how many bananas can be bought for the same overall cost?

• 2 bananas
• 2 bananas
• 3 bananas

The answer is "2 bananas". 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 xx Multiplier = Product

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

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

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

The direct and inverse variations pair.

Multiplicand xx Multiplier = Product

A particular form of rate-formula is common and is repeated in different forms.
rate xx 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.

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

Which of the following is price?

• rate of money per unit
• rate of money per unit
• price is not a rate

The answer is "rate of money per unit".

price xx 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.

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

Which of the following is speed?

• rate of distance per unit time
• rate of distance per unit time
• speed is not a rate

The answer is "rate of distance per unit time".

speed xx 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

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

Which of the following is work-rate?

• rate of work done per unit time
• rate of work done per unit time
• work-rate is not a rate

The answer is "rate of work done per unit time".

work rate xx time rArr 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.

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

Which of the following is fill-rate?

• rate of volume per unit time
• rate of volume per unit time
• fill-rate is not a rate

The answer is "rate of volume per unit time".

fill rate xx time rArr 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

The direct and inverse variations pair.

Multiplicand xx Multiplier = Product

A rate-formula as direct and inverse variation pair.
rate xx span = aggregate.

Various forms of this equation are:
price xx number = cost
speed xx time = distance
work rate xx time rArr work done
fill rate xx time rArr filled amount

slide-show version coming soon