Algebra

Ratio and Proportion

If 𝑎:𝑏∷𝑐:𝑑; then 𝑎𝑑=𝑏𝑐; 𝑎𝑏=𝑐𝑑; 𝑎+𝑏𝑏=𝑐+𝑑𝑑; 𝑎−𝑏𝑏=𝑐−𝑑𝑑; 𝑎+𝑏𝑎−𝑏=𝑐−𝑑𝑑; If 𝑎𝑏=𝑐𝑑=𝑒𝑓; then 𝑎𝑏=𝑎+𝑐+𝑒+⋯𝑏+𝑑+𝑓+⋯

General Theorem

If 𝑎𝑏=𝑐𝑑=𝑒𝑓=⋯=𝑘; then 𝑘=𝑝𝑎𝑛+𝑞𝑐𝑛+𝑟𝑒𝑛+⋯𝑝𝑏𝑛+𝑞𝑑𝑛+𝑟𝑓𝑛+⋯1𝑛 where 𝑝, 𝑞, and, 𝑟 are any quantities whatever.

Proof

Rule: To verify any equation between such proportional quantities: Substitiute for 𝑎, 𝑐, and, 𝑓, theire equivalents 𝑘𝑏, 𝑘𝑑, and, 𝑘𝑓, respectively, in the given equation.

Examples

If 𝑎:𝑏∷𝑐:𝑑; then 𝑎−𝑏𝑐−𝑑=𝑎−𝑏𝑐−𝑑 Put 𝑘𝑏 for 𝑎, and 𝑘𝑑 for 𝑐; thus 𝑎−𝑏𝑐−𝑑=𝑘𝑏−𝑏𝑘𝑑−𝑑=𝑏𝑘−1𝑑𝑘−1=𝑏𝑑; also, 𝑎−𝑏𝑐−𝑑=𝑘𝑏−𝑏𝑘𝑑−𝑑=𝑏𝑘−1𝑑𝑘−1=𝑏𝑑. Identical results being obtained, the proposed equatio must be true.

Continued Proportion

If 𝑎:𝑏:𝑐:𝑑:𝑒:⋯, forming a continued proportion, then 𝑎:𝑐∷𝑎2:𝑏2, the duplicate ratio of 𝑎:𝑏, 𝑎:𝑑∷𝑎3:𝑏3, the triplicate ratio of 𝑎:𝑏, and so on. Also 𝑎:𝑏 is the subduplicate ratio of 𝑎:𝑏, 𝑎32:𝑏32 is the sesquiplicate ratio of 𝑎:𝑏.

Ratio Approach Unity

The fraction 𝑎𝑏 is made to approach nearer to unity in value, by adding the same quantity to the numerator and denominator. Thus 𝑎+𝑥𝑏+𝑥 is nearer to 1 than 𝑎𝑏 is

Compounded Ratios

Def. The ratio compounded of the ratios 𝑎:𝑏 and 𝑐:𝑑 is the ratio 𝑎𝑐:𝑏𝑑 If 𝑎:𝑏∷𝑐:𝑑, and 𝑎':𝑏'∷𝑐':𝑑'; then, by compounding ratios, 𝑎𝑎':𝑏𝑎'∷𝑐𝑐':𝑑𝑑'.

Variation

If 𝑎∝𝑐 and 𝑏∝𝑐, then (𝑎±𝑏)∝𝑐 and 𝑎𝑏∝𝑐. If 𝑎∝𝑏𝑐∝𝑑}, then 𝑎𝑐∝𝑏𝑑 and 𝑎𝑐∝𝑏𝑑. If 𝑎∝𝑏, we may assume 𝑎=𝑚𝑏, where 𝑚 is some constant.

Sources and References

https://archive.org/details/synopsis-of-elementary-results-in-pure-and-applied-mathematics-pdfdrive