Equivalent Equations

Pic of Milk Cartons
Image Source: http://static.guim.co.uk

The above containers are not all equal, but they are “Equivalent”.

They are “equivalent”, because they all contain milk !

In Algebra, we can have “Equivalent” Equations, that look different in size and complexity, but all contain the same number answer for their unknown variable letter.

Equations which have the same solution value are said to be Equivalent.

2N = 6,
2N + 1 = 7
8N = 24
2N/3 = 2
2N – 5 = 1

all have the same solution of N=3.

These five equations are Equivalent.

We can create each of the above equivalent equations, by starting with the basic equation 2N = 6.

Equation 1: 2N + 1 = 7

2N = 6 (+1 both sides)

2N + 1 = 7

Equation 2: 8N = 24

2N = 6 (x 4 both sides)

8N = 24

Equation 3: 2N/3 = 2

2N = 6 (Divide by 3 both sides)

2N/3 = 2

Equation 4: 2N – 5 = 1

2N = 6 ( -5 both sides)

2N – 5 = 1

Equivalent Fraction Equation

A Fraction Equation that has a solution of N=3
is the equation (N + 5) / 2 = 4

Because we are building an equivalent Fraction equation, the mathematics is a little bit more involved.

The fraction equation could be built from our basic starting equation of 2N = 6, by applying the following two operations:
+ 10 both sides, then divide both sides by 4

2N = 6 (+10 both sides)

2N + 10 = 16 (now /4 both sides)

2N/4 + 10/4 = 16/4 (Reduce down the fractions)

n/2 + 5/2 = 4

(N + 5)/2 = 4

Why Make Equivalent Equations ?

Three pieces of Pizza
Image Source: http://www.ajspizzaria.com

Equivalent Equations are useful when we want to “scale up” or “Scale down” a situation.

Let’s have a little “Pizza Party”.

We want each person to have three pieces of Pizza, because this should be the “perfect” amount for them without them over-eating, or going hungry.

Let “N” = the number of pieces per person.

Our solution for each person is N = 3 pieces of Pizza.

How many Pizza pieces do we need for different numbers of people at our little “Pizza Party” ?

If we only have two people, then the equation is 2 x N = 2 x 3 = 6 pieces total.

If we have 8 people, then the equation is 8 x N = 8 x 3 = 24 pieces total.

What if we have two people, but we would like an extra piece left over to give to our older sister (who is on a diet!) .

Then our equation is 2N + 1 = 2 x 3 + 1 = 7 pieces total.

Notice how these situations reflect our previous example Algebra Equations:

2N = 6, 8N = 24, and 2N + 1 = 7.

Equivalent Equations in the Real World

Catering Foods
Image Source: http://www.mycaterer.com.au

Extending our “Pizza Party” example, we could have someone with a Catering business who needs to know how many food items to have avaialble for a function.

For example, they might be supplying “finger food” consisting of mini pizza slices, spring rolls, mini quiches, and so on. For each item they would need to have an estimate of how many pieces to supply for person.

They do not want to supply to little food and leave patrons hungry, but they do not want to have too much food left over at the end that is then wasted.

A set of Equivalent Equations programmed into an Excel Spreadsheet, or even into an iPhone App, could help them work out exactly how much food to provide for 40 people, versus 400 people, and so on.

Something to always keep in mind is it might be useful to add a 10% extra “contingency factor”.

Eg. Have some extra food available to serve, just in case a tray of Spring Rolls all got accidentally burned in the oven, or dropped by a waiter, etc.

Another example of Equivalent Equations might be if you were working for the “World Health Organisation” or the “Red Cross” and need to supply Vaccine to villages and towns to fight a disease epidemic.

China Vacination being given
Image Source: http://www.todaymednews.org

It might be that Children require and injection of 1ml of vaccine, while Adults need 2ml each.

Again a set of equations in a Spreadsheet or computer App could help work out how much Vaccine needs to be taken to a small Village of 20 adults and 8 children, and how much needs to be taken to a Town of 350 Adults, and 100 children.

Any activity involving dividing up items and distributing them could use Equivalent Equations.

Activities where items are allocated to jobs, such as Building, Plumbing, and Electrical work also require scaling up or scaling down depending on the job.

Trades persons can either use some Equivalent Equations mathematics, or else end up with a big shed full of expensive leftovers.

Shed full of lumber timber
Image Source: http://thomassawmill.com

That’s it for Equivalent Equations.

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