A vessel contains a mixture of two liquids, A and B, in the ratio 5 : 3. If 16\text{ litre… — JAMB UTME

A vessel contains a mixture of two liquids, $A$ and $B$ , in the ratio $5 : 3$ . If $16\text{ litres}$ of the mixture is drawn off and replaced with an equal volume of liquid $B$ , the ratio of liquid $A$ to liquid $B$ in the vessel becomes $1 : 1$ . What was the initial total volume of the mixture in the vessel, in litres?

Cevap: 80 litres

Cevap

The initial total volume of the mixture in the vessel was

80\text{ litres}

Let the initial volume of the mixture be

V

litres. Liquid

A

initially comprises

\frac{5}{8}V

litres and liquid

B

comprises

\frac{3}{8}V

litres. When

16\text{ litres}

of mixture is removed, the volume of liquid

A

removed is

\frac{5}{8} \times 16 = 10\text{ litres}

, and the volume of liquid

B

removed is

\frac{3}{8} \times 16 = 6\text{ litres}

. After adding

16\text{ litres}

of pure liquid

B

, the new volume of liquid

A

\frac{5}{8}V - 10

and the new volume of liquid

B

\frac{3}{8}V - 6 + 16 = \frac{3}{8}V + 10

. Since the new ratio is

1 : 1

, setting

\frac{5}{8}V - 10 = \frac{3}{8}V + 10

gives

\frac{2}{8}V = 20

, which simplifies to

V = 80\text{ litres}

Adım Adım Çözüm

Define total initial volume as

V

and express the initial quantities of liquids

A

and

B

Liquid

A = \frac{5}{8}V

litres, Liquid

B = \frac{3}{8}V

litres.

The total ratio parts equal

5 + 3 = 8

Calculate the volume of each liquid removed in

16\text{ litres}

of mixture.

Volume of

A

removed

= 10

litres; Volume of

B

removed

= 6

litres.

The drawn-off mixture retains the original

5 : 3

ratio of liquids.

Formulate expressions for the quantities of

A

and

B

after adding

16\text{ litres}

of pure liquid

B

New amount of

A = \frac{5}{8}V - 10

; New amount of

B = \frac{3}{8}V + 10

16

litres of liquid

B

is added to the remaining quantity of

B

, which was

\frac{3}{8}V - 6

Equate the new quantities of

A

and

B

since the final ratio is

1 : 1

\frac{5}{8}V - 10 = \frac{3}{8}V + 10 \implies \frac{2}{8}V = 20 \implies V = 80

A final ratio of

1 : 1

implies equal quantities of both liquids.

Anahtar Kavram

Ratio modification through mixture removal and replacement

Tahmini Süre:2m 0s

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