Versatile use of Base Equalization Technique by its Abstraction

Base equalization technique has multiple applications

In a problem with multiple parameters, identify the base to be equalized and its target value and equalize the bases. This is base equalizing technique.

We saw how Base equalization technique helped us to solve indices problems and was indispensable in fraction arithmetic. However, Its scope of use is not limited to these two areas only – it is used for joining two ratios as well.

We knew these methods for long. But now we have taken the important step of linking these three applications by stating that these are all in fact applications of the same base equalization technique.

Let us see first how the technique is applied in Ratio problems. Subsequently we will explore briefly the implication of the multiple area of application of the same technique.

Application of base equalization technique in joining ratios

In ratio problems we recognize the use of this technique for joining as well as comparing two or more ratios.

We will explain the application through an example.

Problem example 1

Ratio of ages of A and B is 2 : 3 and between B and C is 4 : 5, while the total age of C and A is 46 years. Age of B in years is,

1. 16
2. 48
3. 24
4. 54

Solution

By analyzing the problem goal we observe that to get B, we need to know the value of either A or C, as in the given ratios B is related to one of A or C. This can easily be done if we can get a second relation between A and C, as one relation, A + C = 46 is already available.

We know that to get values of two unknown variables, two linear equations in the two variables would be sufficient.

So the primary requirement boils down to finding a new relation between A and C.

This can be done by joining the two given ratios and as B is common to both the ratios. We transform the ratios in such a way that B's term values in each ratio equals LCM of its two values, that is, 12.

A second condition for joining two ratios is the common term value to be in the numerator of one ratio and in the denominator in another. This condition is already fulfilled in the given ratios.

Thus transforming the two ratios we get,

$\displaystyle\frac{A}{B} = \displaystyle\frac{2}{3} = \displaystyle\frac{8}{12}$, and

$\displaystyle\frac{B}{C} = \displaystyle\frac{4}{5} = \displaystyle\frac{12}{15}$.

Joining these two ratios through the common term value of B we get the ratio of three quantities as,

$ A : B : C = 8 : 12 : 15$, and between A and C as,

$\displaystyle\frac{A}{C} = \displaystyle\frac{8}{15}$

Or,$\displaystyle\frac{A + C}{C} = \displaystyle\frac{23}{15}$

Or, $C = \displaystyle\frac{46\times{15}}{23} = 30$, and $B = 30\times{\displaystyle\frac{4}{5}} = 24$.

Answer: Option c: 24.

Key concept used: Analyzing the problem to Identify the key requirement as getting a new relation between A and C -- Joining two ratios by base equalization and establish the new relation between A and C -- solving for C from two linear equations -- getting the value of B from the relevant ratio.

Instead of joining if comparison of the two ratios A : B and B : C were required, then also base equalization would have been required but base would have been not the common term B but the denominators of the two fractions representing the two ratios. In that case target equal base value would have been LCM of 3 and 5, that is 15 just as we have done earlier in case of fraction comparison.

The result would have been in that case, A : B = 2 : 3 = 10 : 15 and B : C = 4 : 5 = 12 : 15, that is, B : C > A : B. This operation effectively is a fraction comparison operation and nothing new here.

The new application of the technique is then joining two ratios using base equalization to LCM of the common term in the two ratios, appearing once as denominator and next as numerator.

Thus we find the same Base equalization technique is fruitfully applied in three diverse areas, indices, fractions and now ratios.

But how can the same technique be applied in three different topics?

The reason why this is possible lies in the concept of abstraction. Abstraction means expressing in more general terms. The special characteristic of the base equalization technique is its amenability to abstraction. It is possible to express the common core abstract base equalization technique as a series of general steps,

1. In a set of components each having multiple entities, identify the base to be equalized.
2. Identify the target value to be equalized to.
3. Equalize the bases of each component in the set.
4. Establish direct relationship between the other entities of all components in the set.
5. Carry out the desired operations on these entities that was the objective of the whole series of actions.

Observe the completely general nature of the statements. For each of the three application areas we have identified till now, we need only to convert the general steps to specific as required spawning three appropriate variations of the same technique for three application areas.

This gives rise to the possibility of useful application of the abstract base equalization technique even in far removed areas outside mathematics, such as in happiness comparison.

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