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Substance A has butter to sugar ratio of 8:1 and substance B has butter to sugar ratio of 175:1. The whole of A is combined with part of B to get substance C of Butter to sugar ratio of 30:1. What ratio of substance B was used?

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  • $\begingroup$ Welcome to Math SE! Please add some details to your question about what you've tried and where you're stuck. $\endgroup$
    – DMcMor
    Apr 22, 2019 at 12:46
  • $\begingroup$ Where's the hard part ? I tried to become a pharmacy tech and had to figure these things out before. $\endgroup$
    – user645636
    Apr 22, 2019 at 19:13
  • $\begingroup$ If the ratios are by weight, we need to know the weights of whole A and whole B in order to get an answer. $\endgroup$
    – KY Tang
    Apr 23, 2019 at 2:29
  • $\begingroup$ did the math wrong, but you actually don't have to know how much of A and B, just their relative ratio. alligation is a method useful for this. $\endgroup$
    – user645636
    Apr 23, 2019 at 13:24

2 Answers 2

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I'll look at doing alligation:

$$\begin{array}{c|c|c}\hspace{-1px}11\;\frac{1}{9}&&2\frac{897}{1364}\\\hline&3\frac{7}{31}&\\\hline \frac{25}{44}&&7\frac{247}{279}\end{array}$$

all numbers in the left columns are percentages for the mixtures (8:1 has 9 parts total, 175:1 has 176 parts total). middle column has the wanted ratio ( 30:1 has 31 parts). the last column has parts in the final mixture, turning them to full integers we get them equivalent to 32625 and 96800 reducing to 1305 and 3872 which then works to produce 5177 parts of $\frac{1}{31}$ which is the ratio $30:1$ for the two ingredients.

Probably not the way your teacher wants it done, but it works via subtraction along diagonals, and converting to same denominator fractions on the right then reduction of numerators. Thankfully, I had a calculator to check my work.

technical notes: Assumes percentages are in same units or equivalent units. Doesn't take into account contraction of by-product mixture.

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A has $8x$ butter to $1x$ sugar.

B has $175y$ butter to $1y$ sugar.

C has $30z$ butter to $1z$ sugar.

So

\begin{align} 8x + 175y &= 30z \\ 1x + 1y &= 1z \\ \hline 8x + 175y &= 30z \\ 8x + 8y &= 8z \\ \hline 167y &= 22z \\ y &= \dfrac{2}{17}z \\ x &= \dfrac{15}{17}z \end{align}

We could say

$$\dfrac{15}{17}A + \dfrac{2}{17}B = C$$

To use the whole of $A$, we need to multiply through by $\dfrac{17}{15}$.

$$1A + \dfrac{2}{15}B = \dfrac{17}{15}C$$

So $\dfrac{2}{15}$ of substance $B$ was used.

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