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To make ‘short’ pastry one uses flour to fat in the ratio 2 : 1. To make ‘flaky’ pastry requires a ratio of 4 : 3.

If I have 29 kg of flour and 20 kg of fat and wish to use all the ingredients making some of each type of pastry. How much flaky pastry do I make?

So my approach, long winded no doubt and possibly incorrect results, to make sense of the individual ratios:

  • To make just 'short' pastry, using 29kg of flour + $\frac{29}{2}=14.5kg$ of fat (2:1 ratio), one could make a maximum of 43.5kg of 'short' pastry.

  • To make just 'flaky' pastry, using all the fat this time as 29kg of flour would require 21.75kg of fat i.e. more than we have. 20kg of fat + $20*\frac{4}{3}=26.67kg$ of flour (4:3 ratio), one could make a maximum of 46.67kg of 'flaky' pastry.

How to then "combine" these ratios to make "some of each type of pastry" and use ALL of the ingredients?

Assume to use ALL the ingredients the ratios would have to fit neatly into the given quantities?

Based on @peterwhy direction:

  • To make kg of short pastry we use $\frac{2}{3}s$ of flour $\frac{1}{3}s$ of fat, to make kg of flaky pastry we use $\frac{4}{7}f$ of flour $\frac{3}{7}f$ of fat.

So:

  • total flour used: $\frac{2}{3}s$ + $\frac{4}{7}f$ = 29
  • total fat used: $\frac{1}{3}s$ + $\frac{3}{7}f$ = 20

Solving simultaneously gives: s = 10.5, f = 38.5

So 38.5 kg of 'flaky' pastry is produced using all the ingredients.

Worldly-Guest8115
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    To make $s$ kg of short pastry and $f$ kg of flaky pastry, how much flour do you need? How much fat do you need? Equating what you need and what you have, this will give two linear equations in two variables. – peterwhy Sep 23 '21 at 11:37
  • Ok, so use the resulting quantities 's' and 'f' and equate them to the ingredient ratios we know they will comprise, and the total ingredient quantities we actually have. – Worldly-Guest8115 Sep 23 '21 at 12:45
  • That’s how I would solve this. Now you have an answer, you can post it as an answer down there. https://math.stackexchange.com/help/self-answer – peterwhy Sep 23 '21 at 14:09
  • Thanks very much, much appreciated. – Worldly-Guest8115 Sep 23 '21 at 15:07

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