A chemist has three different acid solutions: Solution X, Solution Y, and Solution Z. In S… — ACT

A chemist has three different acid solutions: Solution $X$ , Solution $Y$ , and Solution $Z$ . In Solution $X$ , the ratio of acid to water is $1:3$ by volume. In Solution $Y$ , the ratio of acid to water is $3:2$ by volume. In Solution $Z$ , the ratio of acid to water is $5:1$ by volume. If the chemist mixes Solution $X$ , Solution $Y$ , and Solution $Z$ in a volume ratio of $4:5:3$ , respectively, what is the ratio of acid to water in the final mixture?

A
$17:7$
B
$34:25$
$13:11$ Cevap
D
$3:2$
E
$101:79$

Cevap

The ratio of acid to water in the final mixture is

13:11

The correct answer is the ratio

13:11

. To find the ratio of acid to water in the final mixture, first convert the individual ratios to part-to-whole fractions: Solution

X

\frac{1}{4}

acid and

\frac{3}{4}

water; Solution

Y

\frac{3}{5}

acid and

\frac{2}{5}

water; and Solution

Z

\frac{5}{6}

acid and

\frac{1}{6}

water. Next, scale these fractions by the volume mixing ratio of

4:5:3

. Assuming

4\text{ units}

X

5\text{ units}

Y

, and

3\text{ units}

Z

, the total acid is

4(\frac{1}{4}) + 5(\frac{3}{5}) + 3(\frac{5}{6}) = 1 + 3 + 2.5 = 6.5\text{ units}

, and the total water is

4(\frac{3}{4}) + 5(\frac{2}{5}) + 3(\frac{1}{6}) = 3 + 2 + 0.5 = 5.5\text{ units}

. The ratio of acid to water is

6.5:5.5

, which simplifies to

13:11

Adım Adım Çözüm

Determine the part-to-whole fractions of acid and water for each solution.

For Solution

X

, the ratio of acid to water is

1:3

, which means the acid fraction is

\frac{1}{4}

and the water fraction is

\frac{3}{4}

. For Solution

Y

, the ratio is

3:2

, meaning the acid fraction is

\frac{3}{5}

and the water fraction is

\frac{2}{5}

. For Solution

Z

, the ratio is

5:1

, meaning the acid fraction is

\frac{5}{6}

and the water fraction is

\frac{1}{6}

Converting part-to-part ratios into fractions of the total volume is necessary to scale each solution correctly.

Choose convenient volumes representing the

4:5:3

mixing ratio, and calculate the volume of acid and water contributed by each solution.

Assume we mix

4\text{ liters}

of Solution

X

5\text{ liters}

of Solution

Y

, and

3\text{ liters}

of Solution

Z

. Acid from

X

4 \times \frac{1}{4} = 1\text{ liter}

; water is

3\text{ liters}

. Acid from

Y

5 \times \frac{3}{5} = 3\text{ liters}

; water is

2\text{ liters}

. Acid from

Z

3 \times \frac{5}{6} = 2.5\text{ liters}

; water is

3 \times \frac{1}{6} = 0.5\text{ liters}

Multiplying the part-to-whole fractions by the respective mixing volumes yields the absolute amounts of acid and water contributed.

Sum the total volumes of acid and water in the final mixture.

Total acid =

1 + 3 + 2.5 = 6.5\text{ liters}

. Total water =

3 + 2 + 0.5 = 5.5\text{ liters}

Finding the total amounts of each component allows us to determine the final composition of the mixture.

Find the simplified integer ratio of total acid to total water.

The ratio of acid to water is

6.5 : 5.5

. Multiplying both terms by

2

to convert to integers gives

13:11

Ratios are standardly expressed as simplified integers.

Anahtar Kavram

Calculating mixture compositions by converting part-to-part ratios to part-to-whole fractions and applying weighted average proportions.

Alternatif Yöntem

Instead of choosing arbitrary volumes like 4, 5, and 3, you can write the total volume as

V

and use algebraic fractions:

\text{Acid} = 4k(\frac{1}{4}) + 5k(\frac{3}{5}) + 3k(\frac{5}{6}) = 6.5k

, and

\text{Water} = 4k(\frac{3}{4}) + 5k(\frac{2}{5}) + 3k(\frac{1}{6}) = 5.5k

. The ratio remains

6.5k : 5.5k = 13:11

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