Question - Complex Mixture Ratio Problem Solving
Solution:
To solve this problem, we need to work step by step, adjusting the mixture according to the changes described, and using the variable $$x$$ to represent the initial volume of the mixture. Let’s break the problem down:1. Initial Mixture Ratio (water to cordial): 7:2.This means that for every 7 parts water, there are 2 parts cordial.2. Adding Cordial to Change Ratio to 4:5.In the initial mixture ratio, the total parts are $$7+2=9$$. If the original volume of the mixture is $$x$$ ml, then we have $$x \cdot \frac{7}{9}$$ ml of water and $$x \cdot \frac{2}{9}$$ ml of cordial. To make the new mixture have a ratio of 4:5, we assume that the amount of water remains constant, while the amount of cordial is increased to make it equal to $$\frac{5}{4}$$ times the amount of water.3. Sam's Mum Takes a 100 ml Sip.Sam's mum takes a sip from the mixture after cordial has been added but before any more water has been added. So, the ratio remains the same, but the volume decreases by 100 ml.4. Diluting the Mixture to a Ratio of 3:1 with Water.After Sam's mum takes the 100 ml sip, the mixture is diluted with water to make a new ratio of 3:1.Now let's perform the calculations step by step:Initial mixture (step 1):- Water: $$x \cdot \frac{7}{9}$$ ml- Cordial: $$x \cdot \frac{2}{9}$$ mlAdditional cordial needed to get 4:5 ratio (step 2):We want the ratio of water to cordial to be 4:5, where the water remains the same. We can set up a proportion to find the new amount of cordial (let's call it $$C$$):\[\frac{4}{5} = \frac{x \cdot \frac{7}{9}}{C}\]Solving for $$C$$:\[C = x \cdot \frac{7}{9} \cdot \frac{5}{4}\]\[C = x \cdot \frac{35}{36}\]The added cordial amount will be the difference between the new and the old cordial amount:\[C_{\text{added}} = C - x \cdot \frac{2}{9} = x \cdot \frac{35}{36} - x \cdot \frac{2}{9}\]After the sip from Sam's mum (step 3), the volume is reduced by 100 ml but the ratio remains 4:5.Final dilution to 3:1 ratio (step 4):Now, the total volume ($$V_F$$) after dilution will be:\[V_F = \text{current volume} + \text{added water}\]The current volume after the sip is:\[V_{\text{current}} = x + C_{\text{added}} - 100\]To get the exact amount of added water, we need to find the volume of cordial after Sam's mum takes her sip, which represents $$\frac{5}{9}$$ of the total mixture then. Let's call this volume $$C_{\text{current}}$$. We solve for $$C_{\text{current}}$$ using the ratio after the sip:\[C_{\text{current}} = \frac{5}{9} \cdot (V_{\text{current}})\]Now the final volume $$V_F$$ should be such that water to cordial ratio is 3:1. Let the added water be $$W_{\text{added}}$$, so:\[W_{\text{added}} + x \cdot \frac{7}{9} = 3 \cdot C_{\text{current}}\]Where $$C_{\text{current}}$$ is the amount of cordial after Sam's mum's sip, which is still $$x \cdot \frac{35}{36} - 100 \cdot \frac{5}{9}$$, since the sip removed 100 ml of the mixture which had 4 parts of water and 5 parts cordial (out of 9 parts total).Combining the equations, you can solve for $$W_{\text{added}}$$ and $$V_F$$.\[V_F = x + (x \cdot \frac{35}{36} - x \cdot \frac{2}{9}) - 100 + W_{\text{added}}\]Substitute $$W_{\text{added}}$$ from the water to cordial ratio equation into the above to get the final volume $$V_F$$ in terms of $$x$$.It's quite a complex problem with several steps, and each step must be handled carefully to ensure the ratios and volumes add up correctly.
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