MSC #8 – Proportionately: Direct Proportion

Elementary school pupils are trained to always reduce a fraction to it lowest terms by dividing both the numerator and denominator by their greatest common factor.

In division having a divisor with decimal parts, the students are also taught to convert the divisor to a whole number by moving the decimal point to the right. The decimal point in the dividend is also shifted by the same number of decimal places. Effectively, both the dividend and the divisor are multiplied by the same power of ten before the division is performed.

It may not be obvious to young learners that the same principle – direct proportion – is at work in both cases.

In our featured example, we did not solve for the distance by the usual method of getting the rate in kilometers per hour and then multiplying it by the number of hours.

Instead, knowing that 7 hours is equivalent to 420 minutes, we used doubling of both the numerator and the denominator twice to easily get the answer.

When the last non-zero digit of the divisor is 5, a simpler solution can be obtained by multiplying both the divisor and the dividend by 2, instead of a power of 10. This “doubling“ can be done several times as needed.

The short cut for dividing by 5 – doubling the dividend then dividing by 10 can be explained by using this technique: 234 ÷ 5 = (234 x 2) ÷ (5 x 2) = 468 ÷ 10 = 46.8

Similarly, the short cut for dividing by 25 is to double the dividend twice then dividing by 100:

234 ÷ 25 = 468 ÷ 50 = 936 ÷ 100 = 9.36

Other examples of this method are:

82.5 ÷ 0.75 = 165 ÷ 1.5 = 330 ÷ 3 = 110.

67.5 ÷ 0.125 = 135 ÷ 0.25 = 270 ÷ 0.5 = 540 ÷ 1.0 = 540