A specialized aerospace component is manufactured from a custom metal alloy weighing exact… — State PSC Exam

A specialized aerospace component is manufactured from a custom metal alloy weighing exactly $160$ kg. The alloy's composition by weight is $0.35$ aluminum and $\frac{5}{16}$ magnesium, with the remaining portion consisting entirely of titanium. What is the exact mass of titanium in this component?

Answer: 54 kg

Answer

To find the mass of titanium, the total proportions of the other metals must first be unified into a common format (either fractions or decimals). By converting the magnesium's share of

\frac{5}{16}

to the decimal

0.3125

, the combined proportion of aluminum and magnesium is

0.35 + 0.3125 = 0.6625

. The titanium makes up the remaining portion of the whole, which is

1 - 0.6625 = 0.3375

. Finally, multiplying this decimal proportion by the total alloy mass of

160

kg yields exactly

54

kg.

Step-by-Step Solution

Convert the fractional part to a decimal.

\frac{5}{16} = 0.3125

Unifying the terms into a single format (decimals) makes addition straightforward.

Add the portions of aluminum and magnesium.

0.35 + 0.3125 = 0.6625

This finds the total proportion of the alloy that is NOT titanium.

Calculate the proportion of titanium.

1 - 0.6625 = 0.3375

The sum of all material proportions in the whole alloy must equal exactly

1

Calculate the final mass of titanium.

0.3375 \times 160 = 54

Multiplying the component's proportion by the total mass gives the specific weight of that component.

Key Concept

Converting between fractions and decimals and applying proportional reasoning to find a part of a whole.

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