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.
ScanSolve AI Answer
Welcome back Sheela — been a while! Let's pick up where you left off. Convert the following fractions: a) To convert (85)/(105) to its simplest form, find the greatest common divisor (GCD) of the numerator and the denominator. Step 1: Find the prime factors of 85 and 105. 85 = 5 × 17 105 = 3 × 5 × 7 Step 2: The GCD of 85 and 105 is 5. Step 3: Divide both the numerator and the denominator by 5. (85 ÷ 5)/(105 ÷ 5) = (17)/(21) The simplified fraction is (17)/(21). b) To convert (-48)/(144) to its simplest form, find the greatest common divisor (GCD) of the numerator and the denominator. Step 1: Find the prime factors of 48 and 144. 48 = 2^4 × 3 144 = 2^4 × 3^2 = 3 × 48 Step 2: The GCD of 48 and 144 is 48. Step 3: Divide both the numerator and the denominator by 48. (-48 ÷ 48)/(144 ÷ 48) = (-1)/(3) The simplified fraction is (-1)/(3). Simplify: a) Simplify 3(1)/(4) - 2(2)/(5) + 1(1)/(10). Step 1: Convert the mixed numbers to improper fractions. 3(1)/(4) = ((3 × 4) + 1)/(4) = (13)/(4) 2(2)/(5) = ((2 × 5) + 2)/(5) = (12)/(5) 1(1)/(10) = ((1 × 10) + 1)/(10) = (11)/(10) Step 2: Rewrite the expression with improper fractions. (13)/(4) - (12)/(5) + (11)/(10) Step 3: Find the least common multiple (LCM) of the denominators 4, 5, and 10. The LCM is 20. Step 4: Convert each fraction to an equivalent fraction with a denominator of 20. (13)/(4) = (13 × 5)/(4 × 5) = (65)/(20) (12)/(5) = (12 × 4)/(5 × 4) = (48)/(20) (11)/(10) = (11 × 2)/(10 × 2) = (22)/(20) Step 5: Perform the addition and subtraction. (65)/(20) - (48)/(20) + (22)/(20) = (65 - 48 + 22)/(20) = (17 + 22)/(20) = (39)/(20) Step 6: Convert the improper fraction back to a mixed number. (39)/(20) = 1(19)/(20) The simplified expression is 1(19)/(20). d) Simplify 5 - 3(1)/(7) + 2(3)/(14). Step 1: Convert the mixed numbers to improper fractions. 3(1)/(7) = ((3 × 7) + 1)/(7) = (22)/(7) 2(3)/(14) = ((2 × 14) + 3)/(14) = (31)/(14) Step 2: Rewrite the expression with improper fractions. 5 - (22)/(7) + (31)/(14) Step 3: Find the least common multiple (LCM) of the denominators 1, 7, and 14. The LCM is 14. Step 4: Convert each term to an equivalent fraction with a denominator of 14. 5 = (5 × 14)/(1 × 14) = (70)/(14) (22)/(7) = (22 × 2)/(7 × 2) = (44)/(14) Step 5: Perform the addition and subtraction. (70)/(14) - (44)/(14) + (31)/(14) = (70 - 44 + 31)/(14) = (26 + 31)/(14) = (57)/(14) Step 6: Convert the improper fraction back to a mixed number. (57)/(14) = 4(1)/(14) The simplified expression is 4(1)/(14). Compare the fractions: a) Compare (5)/(8) and (4)/(7). Step 1: Find a common denominator for 8 and 7. The LCM is 8 × 7 = 56. Step 2: Convert both fractions to equivalent fractions with a denominator of 56. (5)/(8) = (5 × 7)/(8 × 7) = (35)/(56) (4)/(7) = (4 × 8)/(7 × 8) = (32)/(56) Step 3: Compare the numerators. Since 35 > 32, it follows that (35)/(56) > (32)/(56). Therefore, (5)/(8) > (4)/(7). e) Compare (-4)/(19) and (-3)/(17). Step 1: Find a common denominator for 19 and 17. The LCM is 19 × 17 = 323. Step 2: Convert both fractions to equivalent fractions with a denominator of 323. (-4)/(19) = (-4 × 17)/(19 × 17) = (-68)/(323) (-3)/(17) = (-3 × 19)/(17 × 19) = (-57)/(323) Step 3: Compare the numerators. For negative numbers, the number closer to zero is greater. Since -68 < -57, it follows that (-68)/(323) < (-57)/(323). Therefore, (-4)/(19) < (-3)/(17). Got more? Send 'em 📸