A container holds a mixture of sand, gravel, and cement. By weight, the mixture is 30\% sa… — ACT

A container holds a mixture of sand, gravel, and cement. By weight, the mixture is $30\%$ sand, $\frac{2}{5}$ gravel, and the remaining portion is cement. To adjust the mixture's properties, the amount of sand is doubled, the amount of gravel is increased by $25\%$ , and the amount of cement is decreased by $50\%$ . What percentage of the new mixture, by weight, is cement?

$12\%$ Answer
B
$15\%$
C
$20\%$
D
$33\frac{1}{3}\%$
E
$40\%$

Answer

12%

To find the correct percentage, we establish that out of an initial

100\text{ g}

of the mixture, there are

30\text{ g}

of sand,

40\text{ g}

of gravel, and

30\text{ g}

of cement. After the changes, the new weights are

60\text{ g}

of sand,

50\text{ g}

of gravel, and

15\text{ g}

of cement. The new total weight is

125\text{ g}

. Thus, cement constitutes

\frac{15}{125} = 12\%

of the new mixture.

Step-by-Step Solution

Determine the initial proportions and assign a hypothetical total weight of

100\text{ g}

to simplify the calculations.

Sand is

30\text{ g}

. Gravel is

\frac{2}{5} \times 100\text{ g} = 40\text{ g}

. Cement is the remaining weight:

100\text{ g} - (30\text{ g} + 40\text{ g}) = 30\text{ g}

Establishing concrete weights makes applying percentage changes straightforward.

Apply the specified weight changes to each component to find their new weights.

New sand weight:

30\text{ g} \times 2 = 60\text{ g}

. New gravel weight:

40\text{ g} \times 1.25 = 50\text{ g}

. New cement weight:

30\text{ g} \times 0.50 = 15\text{ g}

This calculates the individual component weights after the adjustments.

Calculate the new total weight of the mixture and the new percentage of cement.

New total weight:

60\text{ g} + 50\text{ g} + 15\text{ g} = 125\text{ g}

. Cement percentage:

\frac{15\text{ g}}{125\text{ g}} \times 100\% = 12\%

The percentage of a component in a mixture is its weight divided by the new total weight of the mixture.

Key Concept

Calculating new percentage concentrations after changes in individual component weights within a mixture.

Estimated Time:2m 0s

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