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question 64
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reduce the following to its lowest terms
\\(\frac{5,100}{3,600} = a \frac{x}{y}\\)
solve for a in the above question;
solve for x in the above question;
solve for y in the above question;
question 65
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reduce the following to its lowest terms
\\(\frac{39}{42} = \frac{x}{y}\\)
solve for x in the above question;
solve for y in the above question;

Explanation:

Question 64

Step 1: Find the greatest common divisor (GCD) of 5100 and 3600.

First, factorize both numbers:

• \(5100 = 2^2 \times 3 \times 5^2 \times 17\)
• \(3600 = 2^4 \times 3^2 \times 5^2\)

The GCD is the product of the lowest powers of common prime factors: \(2^2 \times 3 \times 5^2 = 4 \times 3 \times 25 = 300\)

Step 2: Divide numerator and denominator by GCD.

Divide 5100 by 300: \(5100 \div 300 = 17\)
Divide 3600 by 300: \(3600 \div 300 = 12\)

So, \(\frac{5100}{3600} = 1 \frac{5}{12}\) (Wait, actually, when reducing to lowest terms as a fraction, it's \(\frac{17}{12}\), but if we consider the mixed number, \(5100 \div 3600 = 1\) with a remainder. Wait, no, \(5100 = 3600 \times 1 + 1500\), then \(3600 = 1500 \times 2 + 600\), \(1500 = 600 \times 2 + 300\), \(600 = 300 \times 2 + 0\), so GCD is 300. Then \(\frac{5100}{3600} = \frac{5100 \div 300}{3600 \div 300} = \frac{17}{12}\). Wait, maybe the problem has a typo, 5,100 is 5100, 3,600 is 3600. So \(\frac{5100}{3600} = \frac{17}{12}\), which is \(1 \frac{5}{12}\). So if the form is \(A \frac{x}{y}\), then:

• \(A\) is the whole number part, \(x\) is the numerator of the fraction, \(y\) is the denominator.

Step 3: Solve for A, x, y.

\(\frac{5100}{3600} = 1 \frac{5}{12}\) (since \(17 \div 12 = 1\) with remainder 5, so \(17 = 12 \times 1 + 5\), so \(\frac{17}{12} = 1 + \frac{5}{12} = 1 \frac{5}{12}\))

So:

• \(A = 1\)
• \(x = 5\)
• \(y = 12\)

Question 65

Step 1: Find GCD of 39 and 42.

Factorize:

• \(39 = 3 \times 13\)
• \(42 = 2 \times 3 \times 7\)

GCD is 3.

Step 2: Divide numerator and denominator by GCD.

\(39 \div 3 = 13\)
\(42 \div 3 = 14\)

So \(\frac{39}{42} = \frac{13}{14}\), so \(x = 13\), \(y = 14\)

Answer:

(Question 64):

• Solve for A: \(1\)
• Solve for X: \(5\)
• Solve for Y: \(12\)