A wine company needs to blend a California wine with a 5%...

A wine company needs to blend a California wine with a 5$\%$ alcohol content and a French wine with a 9$\%$ alcohol content to obtain 200 gallons of wine with a 7$\%$ alcohol content. How many gallons of each kind of wine must be used?

Key Concepts

Mixture Problems

Mixture problems involve combining two or more substances with differing characteristics to achieve a final mixture with a desired property. This often requires setting up equations that balance quantities like volume and concentration to find the correct proportions of each substance.

System of Linear Equations

When solving mixture problems, one typically formulates a system of linear equations that represents the relationships between the quantities being mixed. These equations account for total volume and the weighted contributions of characteristics such as concentration, and they are solved simultaneously to find the unknown amounts.

Weighted Averages

The concept of weighted averages is crucial in mixture problems because the overall property of the final mixture is determined by the weighted contributions of each component. Each individual component's value (such as percentage concentration) contributes in proportion to its amount in the mixture, leading to an average that reflects the desired final specification.

Transcript

00:01 A wine company needs to blend a californian wine with 5 % alcohol content and a fridge wine with 9 % alcohol content to obtain 200 gallons of wine with a 7 % alcohol content.

00:17 How many gallons of each kind must be used? so when we are tackling this kind of person, the first process is to create variables.

00:30 Variables we get them from the question itself.

00:34 So the question says how many gallons of each kind of wine must be used.

00:42 So we will say let the canons first step, let the gallons, the gallons of california wine equals to x is equal to x.

01:09 All right, list gallons of californian wine that is the 5 december and let the gallons of french wine equals to why i will go back to our question and the question tells us that if we mix we want to obtain 200 gallons of wine all in all so in terms of the gallons of wine, we want 200 gallons.

01:54 So we are saying here that if we mix gallons of california wine plus the gallons of french wine, we should be able to get 200 gallons.

02:11 The question is at what proportion of these 200 gallons are we supposed to mix? so that's the first set.

02:23 Then the second one, we are also told that the californian wine is 5 % alcohol content and the french wine is 9 % alcohol content and we must obtain 7 % alcohol content.

02:42 So we can say for the californian which we say is x, we want 5 % so.

02:51 0 .05 of x plus 0 .09 y, which is french wine, should give us, how many percentages, should give us 7 % of alcohol content in our 200 gallons.

03:29 So it should give us 0 .07 of 200.

03:41 We can basically simplify this by just multiplying 200 by 0 .07.

03:49 We can say we can just go straight into the calculator.

03:55 Multiplying this two, we get 14.

03:59 So we can say this will give us 14, whereby 14 is 0 .0 7 by 200.

04:18 That's how we are getting the 14.

04:23 Now once we have our system, the next step is just to solve...