This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions to the problems: 4. Find the GCF of 40 and 120. Step 1: Find the prime factorization of 40. 40 = 2 × 20 = 2 × 2 × 10 = 2 × 2 × 2 × 5 = 2^3 × 5^1 Step 2: Find the prime factorization of 120. 120 = 2 × 60 = 2 × 2 × 30 = 2 × 2 × 2 × 15 = 2 × 2 × 2 × 3 × 5 = 2^3 × 3^1 × 5^1 Step 3: Identify the common prime factors and their lowest powers. The common prime factors are 2 and 5. The lowest power of 2 is 2^3. The lowest power of 5 is 5^1. Step 4: Multiply the common prime factors with their lowest powers. GCF(40, 120) = 2^3 × 5^1 = 8 × 5 = 40 Answer: The GCF of 40 and 120 is 40. 5. Write the decimal number 0.8333333... as a rational number. Step 1: Let x equal the decimal. x = 0.8333... Step 2: Multiply by 10 to move the non-repeating part to the left of the decimal. 10x = 8.333... (1) Step 3: Multiply by 100 to move one full repeating block to the left of the decimal. 100x = 83.333... (2) Step 4: Subtract equation (1) from equation (2) to eliminate the repeating part. 100x - 10x = 83.333... - 8.333... 90x = 75 Step 5: Solve for x and simplify the fraction. x = (75)/(90) Divide both the numerator and the denominator by their greatest common divisor, which is 15. x = (75 ÷ 15)/(90 ÷ 15) = (5)/(6) Answer: The rational number is (5)/(6). 6. Complete the equivalent fraction by filling in the box. (19)/(7) = (76)/() Step 1: Determine the factor by which the numerator 19 was multiplied to get 76. 19 × k = 76 k = (76)/(19) = 4 Step 2: Multiply the denominator 7 by the same factor. 7 × 4 = 28 Answer: The completed equivalent fraction is (19)/(7) = (76)/(28). 7. Express 16/3 as a decimal correct to 4 decimal places. Step 1: Divide 16 by 3. (16)/(3) = 5.333333... Step 2: Identify the first 5 decimal places to determine rounding. The fifth decimal place is 3. Step 3: Round to 4 decimal places. Since the fifth decimal place (3) is less than 5, we round down (keep the fourth decimal place as it is). 5.3333 Answer: 16/3 as a decimal correct to 4 decimal places is 5.3333. That's 2 down. 3 left today — send the next one.