Question

A chef is using a mixture of two brands of Italian dressing. The first brand contains 10% vinegar, and the second brand contains 13% vinegar. The chef wants to make 210 milliliters of a dressing that is 12% vinegar. How much of each brand should she use?

(I've read like 4 different ways to do this and for some reason I can't wrap my head around it)

Answer 1

Let the required quantity of the 10% vinegar dressing be a (ml), and let the required quantity of the 13% vinegar be b (ml).
0.1a + 0.13b = 210 * 0.12 ...................(1)
a + b = 210 ................................(2)
Now we can rearrange equation (2) to give b in terms of a as follows:
b = 210 - a .................................(3)
Next we can substitute the value of b given in equation (3) into equation (1) giving:
0.1a + 0.13(210 - a) = 210 * 0.12
which simplifies to:
0.1a + 27.3 - 0.13a = 25.2 .........(4)
Now all that is needed is to solve equation (4) to find the value of a. Can you do that?

Answer 2

I think my issue is that I'm not even understanding how everyone is getting from 1 to 2 on this. How do you get a+b=210 from 0.1a + 0.13b=210*0.12? Aren't the percentages need there?

Answer 3

The percentages are put into decimal form in equation (1).
0.1a + 0.13b = 210 * 0.12 ...................(1)
We have a (ml) of 10% vinegar dressing. Therefore the amount of vinegar in a (ml) of dressing is 0.1a.
We have b (ml) of 13% vinegar dressing. Therefore the amount of vinegar in b (ml) of dressing is 0.13b
We have 210 ml of 12% vinegar dressing. Therefore the amount of vinegar in 210 ml of the final dressing is 210 multiplied by 0.12.

a + b = 210 ................................(2)
We let 'a' be the required amount of 10% vinegar dressing and 'b' be the required amount of 13% vinegar dressing. Therefore when a and b are added they must make 210 ml.

Answer 4

A=70, B=140?

Answer 5

We have two unknown quantities in this question:
1. The required amount of 10% vinegar dressing.
2. The required amount of 13% vinegar dressing.
Therefore we need to create a pair of simultaneous equations to be able to solve to find the values.

Answer 6

am i even close..? lol

Answer 7

Yes, your values for a and b are correct.
70 ml and 140 are the total amounts of the two original types of dressing. The amounts of vinegar in each are:
70 * 0.1 = 7 ml
and
140 * 0.13 = 18.2 ml

Answer 8

The amount of vinegar in the final dressing is:
210 * 0.12 = 25.2 ml
which comes from:
7 + 18.2 = 25.2 ml