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3 years ago

Solve the problem by defining the unknown variable, show any formula used, set up an

Mathematics

1 answer:

ivann1987 [24]3 years ago

8 0

Answer:

11, 43, 80

Step-by-step explanation:

A second number is 10 more than three times a first number

The third number is 14 more than six times the first number

The sum of the three numbers is 134. What are the three

numbers?

call the first number " x "

call the second number " y "

call the third number " z "

now represent the given information algebraically

y - 10 = 3x

manipulate for the problem:

y = 3x +10

z - 14 = 6x

manipulate for the problem:

z = 6x +14

x + y + z =134

now use substitution to form one equation

x + y + z = 134

x + 3x + 10 + 6x + 14 = 134

simplify and solve by inverse operations:

10x + 24 = 134

10x = 110

x = 11

now substitute back into to find the answer to the problem

y= 3x + 10

y = 3 * 11 +10

y = 43

z = 6x + 14

z = 6 * 11 + 14

z = 80

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Given that the jeweler has alloys that are 25% gold, 50% gold, and 82% gold.

As he wants to make 14 grams of an alloy by adding two different alloys that is precisely 75% gold, so one alloy must have a percentage of gold more than 75%.

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Let x grams of 82% gold and y grams of 50% gold added to make x+y=14 grams of 75% gold, so

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$\Rightarrow 75/100 \times 14 = 82/100 \times x + 50/100 \times (14-x)$ [as x+y=14]

$\Rightarrow 75 \times 14 = 82 \times x + 50 \times (14-x) \\\\\Rightarrow 75 \times 14 = 82 \times x + 50 \times14-50\times x \\\\\Rightarrow 75 \times 14 = 32 \times x + 50 \times14 \\\\\Rightarrow 32 \times x =75 \times 14 - 50 \times14 \\\\$

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and $y = 14-x= 14-10.9375=3.0625$ grams.

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Case 2: 82% gold + 25% gold

Let x grams of 82% gold and y grams of 25% gold added to make x+y=14 grams of 75% gold, so

75% of 14 = 82% of x + 25% of y

$\Rightarrow 75/100 \times 14 = 82/100 \times x + 25/100 \times y \\\\\Rightarrow 75/100 \times 14 = 82/100 \times x + 25/100 \times (14-x) \\\\ \Rightarrow 75 \times 14 = 82 \times x + 25 \times (14-x) \\\\\Rightarrow 75 \times 14 = 82 \times x + 25 \times14-25\times x \\\\\Rightarrow 75 \times 14 = 57 \times x + 25 \times14 \\\\\Rightarrow 57 \times x =75 \times 14 - 25 \times14 \\\\$

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and $y = 14-x= 14-12.28=1.72$ grams.

Hence, 12.28 grams of 82% gold and 1.72 grams of 50% gold added to make 14 grams of 75% gold.

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