Direct Proportion

As one value increases, so does the other.

Unitary Method

1. Find the value of 1 unit
2. Multiply to find the required number of units

Example

4 books cost £1.28. How much do 7 books cost?

\[ 4 \text{ books} = £1.28 \] \[ 1 \text{ book} = \frac{1.28}{4} = £0.32 \] \[ 7 \text{ books} = 7 \times 0.32 = £2.24 \]

Vedic Method

Using a ratio equation:

\[ 4 : 1.28 \; :: \; 7 : x \]

In Vedic maths, we use the idea that the product of the means equals the product of the extremes.

\[ 4 : 1.28 \; :: \; 7 : x \] \[ 4x = 1.28 \times 7 \] \[ 4x = 8.96 \] \[ x = \frac{8.96}{4} = 2.24 \]

So 7 books cost £2.24.

By the Rule of Three

The Rule of Three is another way to solve direct proportion problems.

\[ 4 \text{ books} = £1.28 \] \[ 7 \text{ books} = x \] \[ \frac{x}{7} = \frac{1.28}{4} \] \[ x = 7 \times \frac{1.28}{4} \] \[ x = 7 \times 0.32 = £2.24 \]

This is really just the same as finding a scale factor for the cost of the books:

\[ \text{Scale factor} = \frac{7}{4} \] \[ x = 1.28 \times \frac{7}{4} \] \[ x = 1.28 \times 1.75 = £2.24 \]

Indirect Proportion

As one value increases, the other decreases.

Example

If it takes 4 days for 10 men to dig a trench, how long will it take 8 men?

\[ \text{More men} \;\Rightarrow\; \text{fewer days} \] \[ \text{This is indirect proportion.} \] \[ 4 \text{ days} \times 10 \text{ men} = x \text{ days} \times 8 \text{ men} \] \[ 40 = 8x \] \[ x = \frac{40}{8} = 5 \]

It will take 5 days.

Vedic Method

“The first by the first and the last by the last.”

For indirect proportion:

\[ 4 : 10 \; :: \; 8 : x \]

\[ 4 : 10 \; :: \; 8 : x \] \[ 4x = 10 \times 8 \] \[ 4x = 80 \] \[ x = \frac{80}{4} = 20 \]

So it will take 20 men‑days (equivalent to 5 days for 8 men).

By the Rule of Three

Since this is indirect proportion, the inverse applies.

\[ 4 \text{ days} \leftrightarrow 10 \text{ men} \] \[ x \text{ days} \leftrightarrow 8 \text{ men} \] \[ \text{Indirect proportion: } \frac{x}{4} = \frac{10}{8} \] \[ x = 4 \times \frac{10}{8} \] \[ x = 4 \times \frac{5}{4} \] \[ x = 5 \]

It will take 5 days.