Diamond and Gold Ring Values

The price of a diamond varies as the square of its weight. Three rings of equal weight, each composed of a diamond set in gold, have values $\pounds a,\pounds b,\pounds c$, the diamonds in them weighing 3, 4, 5 carats respectively. Shew that the value of a diamond of one carat is $\pounds \left(\frac{a+c}{2}-b\right)$, the cost of workmanship being the same for each ring.

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. III. Variation, Examples III, No. 19