Math help please?

Question

1. A chemist has 20 milliliter a of a solution that is 40% acid. She wants to increase the acid content of the solution to make it a 50% acid solution. How many milliliters of pure acid should she add to the solution?

2. The sum of two numbers is 18. Twice the smaller number decreased by 3 equals the large number. What are the two numbers?

3. 60 is 20% of what number?

4. A metal alloy weighing 8 lb and containing 45% of platinum is melted and mixed with 2 lb of a different alloy which contains 50% platinum. What percent if the resulting alloy is platinum?

Answer 1

(2).

Substitution Method :

Let x be the small number and y be the large number.

The word " the sum of two numbers x and y is 18 " represent the equation is x + y = 18.

The word " Twice the smaller number x decreased by 3 equals the large number y " represent the equation is 2x - 3 = y.

The system of equations are x + y = 18 and 2x - 3 = y.

Step 1 : The equation 2 : y = 2x - 3 is already solved for y.

Step 2 : Substitute 2x - 3 for y in the equation 1 : x + y = 18 to find the value of x.

x + (2x - 3) = 18

3x - 3 = 18

Add 3 to each side.

3x - 3 + 3 = 18 + 3

3x = 21

Divide each side by 3.

3x/3 = 21/3

x = 7.

 

Step 3 : Substitute 7 for x in either equation to find y.

Equation 2 : y = 2x - 3

y = 2(7) - 3

y = 14 - 3

y = 11.

 

The solution (x, y) = (7, 11).

Answer 2

(3).

Let x be the any number.

The word " 60 is 20% of what number x " represent the equation is 60 = (20 %)(x).

The above equation solve for x.

60 = 20x/100

60 = x/5

Multiply each side by 5.

60 * 5 = (x/5) * 5

300 = x.

The solution x = 300.

Answer 3

(1).

The volume of the acid solution = 20 milliliter

The percent of acid in 20 ml solution is 40.

The volume of the pure acid in the acid solution = (40%) * 20

                                                                         = (40/100) * 20

                                                                         = 8 ml.

Let us assume that x is a volume of pure acid in milliliters, which should be added to 20 ml of the solution to get a 50% solution.

After addition of x ml  acid, the volume of the acid solution = 20 ml + x ml = (20 + x) ml.

After addition of x ml acid, the volume of pure acid = 8 ml + x ml = (8 + x) ml (milliliters).

Since the percent of acid after adding pure acid should be equal to 50%, this gives us an equation

8 + x  = 50%(20 + x)

8 + x  = (1/2)(20 + x)

2(8 + x)  = (20 + x)

Apply distributive property : a(b + c) = ab + ac.

2(8) + 2(x) = (20 + x)

16 + 2x  = 20 + x

Subtract x from each side.

16 + 2x - x  = 20 + x - x

16 + x = 20

Subtract 16 from each side.

16 + x -16  = 20 - 16

x = 4 ml.

4 ml of the pure acid should be added to 20 milliliters of the 40% acid solution to get the 50% acid solution.

 

Answer 4

(4).

The amount of platinum in metal alloy = 8 lb * 45%

                                                           = 8 * (45/100)

                                                           = 3.6 lb

The amount of platinum in other alloy = 2 lb * 50%

                                                          = 2 * (50/100)

                                                          = 1 lb

The total amount of alloy = 8lb + 2lb

                                       = 10 lb.

The total amount of platinum = 3.6 lb + 1lb

                                              = 4.6 lb.

Percentage of Platinum = (4.6 lb/10 lb ) * 100

                                     = 46%.