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Two mixtures A and B contain milk and water in the ratios [#permalink]

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02 Jan 2008, 07:16

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Two mixtures A and B contain milk and water in the ratios 2:5 and 5:4 respectively. How many gallons of A must be mixed with 90 gallons of B so that the resultant mixture contains 40% milk?

Two mixtures A and B contain milk and water in the ratios 2:5 and 5:4 respectively. How many gallons of A must be mixed with 90 gallons of B so that the resultant mixture contains 40% milk?

How can I do this without using the ratio approach?

As you may have seen in a previous post, I like to shy away from algebra if there is some other framework I can use instead. For mixtures and solutions, I simply draw this box, fill in what is given, and then make everything work out to the bottom right corner.

The rows represent the solutions being mixed. The first solution first, the added solution second, and the resulting solution third. The columns represent the parts of the solution. The first (left) column is the total solution, the middle represents the percentage of the "stuff" (in this case, milk, but could be acid, salt, alcohol, etc), and the third column is the total "stuff" in the solution.

We multiply the first column by the second column to get the third column. When we mix them together, the total amounts mix, as do the amounts of the "stuff", so we add down. Note that the middle column doesn't add - it's just there to get us from the left to the right.

So here is how I applied it to this problem. We know how much of B there is, and we are asked how much of A is mixed with it to get 40% milk all together. Follow the chart, and you see that the bottom row multiplies to the right box, and the right column adds to the right box, so we have the right box from two directions. That's how we solve.

50 + 2/7x = .4(90 + x) x = 122.5

Check out some earlier posts I made on this topic.

Mixture B is a 5:4 ratio mixture of Milk:Water. In other words, for every 5 gallons of milk in B there are 4 gallons of water. To find out the actual number of gallons of milk in X gallons of mixture B there are a couple ways to go about it:

If we have 5 gallons of milk there are 4 gallons of water. So milk = 5/(5+4) or 5/9 is milk. You add 4 and 5 together to get the total volume of the mixture (milk + water) 90 gallons of mixture * 5/9 milk = 50 gallons of milk.

OR

You can see that 5:4 add up to 9 and that 90 is a multiple of 9. If 5:4 = 9 and 9*10 = 90 and we have 90 gallons of the mixture just multiply the ratio by 10. 5*10:4*10 = 50:40 = 50 gallons of milk

Let's say we have milk:water in a 1:9 ratio. We have 15 gallons of the mixture, how many gallons of milk?

1+9=10 so 1 gallon of milk for every 10 gallons of mixture 15/10 = 1.5 1*1.5:9*1.5 multiply each side by (15/10 or 1.5) 1.5:13.5 1.5+13.5 = 15 and 1.5 gallons of it is milk.

Two mixtures A and B contain milk and water in the ratios 2:5 and 5:4 respectively. How many gallons of A must be mixed with 90 gallons of B so that the resultant mixture contains 40% milk?

A. 144 B. 122.5 C. 105.10 D. 72 E. 134

When solving some mixture questions, I find it useful to sketch the solutions with the ingredients SEPARATED.

First recognize that if mixture A has a milk to water ratio of 2:5, then the mixture is 2/7 milk. Also recognize that if mixture B has a milk to water ratio of 5:4, then the mixture is 5/9 milk.

Start with 90 gallons of mixture B, which is 5/9 milk: When we draw this with the ingredients separated, we see we have 50 gallons of milk in the mixture.

Next, we'll let x = the number of gallons of mixture A we need to add. Since 2/7 of mixture A is milk, we know that (2/7)x = the volume of MILK in this mixture:

At this point, we can ADD the two solutions (PART BY PART) to get the following volumes:

Since the RESULTING mixture is 40% milk (i.e., 40/100 of the mixture is milk), we can write the following equation: [50 + (2/7)x]/(90 + x) = 40/100 Simplify to get: [50 + (2/7)x]/(90 + x) = 2/5 Cross multiply to get: 5[50 + (2/7)x] = 2(90 + x) Expand: 250 + (10/7)x = 180 + 2x Subtract 180 from both sides to get: 70 + (10/7)x = 2x Multiply both sides by 7 to get: 490 + 10x = 14x Rearrange: 490 = 4x Solve: x = 490/4 = 245/2 = 122.5

yea, figured it out. what u did was u took the total amount/total ratio = multiplier, in turn usnig the multilpier to find the respective amounts in the ratio.

Are we missing the final step? 2/7 x = 35
_________________

You tried your best and you failed miserably. The lesson is 'never try'. -Homer Simpson

Nope, we need 122.5 gallons of mixture A to be mixed in with mixture B to have 40% milk. The question is just asking how much A we need to mix in, but 35 gallons is indeed the amount of milk in the 122.5 gallons of Mixture A.

we're given that B has a total of 90 gallons. My first step is to figure out how much milk is in 90 gallons of B. Given the ratio 5:4 milk:water, i translate this into 5:9 milk:total.

90 gallons has 50 gallons milk and 40 water.

Now, since we are given info about milk, I put all numbers in terms of milk ... so the ratio of A goes from 2:5 milk:water to 2:7 milk:total. We want to find out how much of A we are going to add, so I set up as follows:

(50+2/7x) / (90+x) = 0.4 (or 40/100 or 4/10, whatever you want). X is the amount of A we will be adding.

I think, the easiest way to resolve this type of questions is making a simple table Component/Quantity of milk-water and then constructing an equation. For component A we will have 2/7A gallons of milk and 5/7 gallons of water. For component B we will have 5/9 gallons of milk and 4/9 of water. Therefore, we will have that:

0.4=(2/7A+5/9B)/(A+B) where B=90. You solve the equation and obtain A=35.

I think, the easiest way to resolve this type of questions is making a simple table Component/Quantity of milk-water and then constructing an equation. For component A we will have 2/7A gallons of milk and 5/7 gallons of water. For component B we will have 5/9 gallons of milk and 4/9 of water. Therefore, we will have that:

0.4=(2/7A+5/9B)/(A+B) where B=90. You solve the equation and obtain A=35.

but that answers the question of how much milk must be added, we are solving for how much of mixture A must be added. you can still get to the answer, it's just an extra step

I think, the easiest way to resolve this type of questions is making a simple table Component/Quantity of milk-water and then constructing an equation. For component A we will have 2/7A gallons of milk and 5/7 gallons of water. For component B we will have 5/9 gallons of milk and 4/9 of water. Therefore, we will have that:

0.4=(2/7A+5/9B)/(A+B) where B=90. You solve the equation and obtain A=35.

but that answers the question of how much milk must be added, we are solving for how much of mixture A must be added. you can still get to the answer, it's just an extra step

The question is about how much quantity of mixture A has to be added (".....How many gallons of A must be mixed with 90 gallons....") and that is what I have calculated.

I think, the easiest way to resolve this type of questions is making a simple table Component/Quantity of milk-water and then constructing an equation. For component A we will have 2/7A gallons of milk and 5/7 gallons of water. For component B we will have 5/9 gallons of milk and 4/9 of water. Therefore, we will have that:

0.4=(2/7A+5/9B)/(A+B) where B=90. You solve the equation and obtain A=35.

but that answers the question of how much milk must be added, we are solving for how much of mixture A must be added. you can still get to the answer, it's just an extra step

The question is about how much quantity of mixture A has to be added (".....How many gallons of A must be mixed with 90 gallons....") and that is what I have calculated.

0.4=(2/7A+5/9B)/(A+B) where B=90. You solve the equation and obtain A=35.

so you did set it up to get mixture A as your answer, but somewhere in your calculations you mixed up and found the amount of milk in mixture A instead (122.5*2/7) = 35.

Check your answer 35 to see for yourself ((90*5/9)+(35*2/7))/90+35 (50+10)/125 60/125 = 48%

Now try it with 122.5 ((90*5/9)+(122.5*2/7)/90+122.5 (50+35)/212.5 85/212.5 = 40%

I think, the easiest way to resolve this type of questions is making a simple table Component/Quantity of milk-water and then constructing an equation. For component A we will have 2/7A gallons of milk and 5/7 gallons of water. For component B we will have 5/9 gallons of milk and 4/9 of water. Therefore, we will have that:

0.4=(2/7A+5/9B)/(A+B) where B=90. You solve the equation and obtain A=35.

Sorry, I was wrong.... A=122.5; I do not know where I got lost

I think, the easiest way to resolve this type of questions is making a simple table Component/Quantity of milk-water and then constructing an equation. For component A we will have 2/7A gallons of milk and 5/7 gallons of water. For component B we will have 5/9 gallons of milk and 4/9 of water. Therefore, we will have that:

0.4=(2/7A+5/9B)/(A+B) where B=90. You solve the equation and obtain A=35.

Sorry, I was wrong.... A=122.5; I do not know where I got lost

No worries, your equation was set up perfectly. Try working it through again and see if you can spot your mistake. If you still can't find it you're welcome to post it up here and we'll be happy to take a look at it for you.

No of gallon milk from A+No of gallon milk from B = (total gallon of mixture)*40/100(40% is milk in tat) Since we are comparing only the milk so both RHS and LHS should be milk.

I found Ian7777's all posts very clear, detail and useful. More than that his attitude is humble and highly positive. I always enjoy reading his posts and learned a lot from him.

Thanks Ian.

GT

ian7777 wrote:

bmwhype2 wrote:

Two mixtures A and B contain milk and water in the ratios 2:5 and 5:4 respectively. How many gallons of A must be mixed with 90 gallons of B so that the resultant mixture contains 40% milk?

How can I do this without using the ratio approach?

As you may have seen in a previous post, I like to shy away from algebra if there is some other framework I can use instead. For mixtures and solutions, I simply draw this box, fill in what is given, and then make everything work out to the bottom right corner.

The rows represent the solutions being mixed. The first solution first, the added solution second, and the resulting solution third. The columns represent the parts of the solution. The first (left) column is the total solution, the middle represents the percentage of the "stuff" (in this case, milk, but could be acid, salt, alcohol, etc), and the third column is the total "stuff" in the solution.

We multiply the first column by the second column to get the third column. When we mix them together, the total amounts mix, as do the amounts of the "stuff", so we add down. Note that the middle column doesn't add - it's just there to get us from the left to the right.

So here is how I applied it to this problem. We know how much of B there is, and we are asked how much of A is mixed with it to get 40% milk all together. Follow the chart, and you see that the bottom row multiplies to the right box, and the right column adds to the right box, so we have the right box from two directions. That's how we solve.

50 + 2/7x = .4(90 + x) x = 122.5

Check out some earlier posts I made on this topic.

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