simplify 7/12+3/8-1/20

{ "query": { "display": "simplify $$\\frac{7}{12}+\\frac{3}{8}-\\frac{1}{20}$$", "symbolab_question": "SIMPLIFY#simplify \\frac{7}{12}+\\frac{3}{8}-\\frac{1}{20}" }, "solution": { "level": "PERFORMED", "subject": "Algebra", "topic": "Algebra", "subTopic": "Simplify", "default": "\\frac{109}{120}", "decimal": "0.90833…", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$\\frac{7}{12}+\\frac{3}{8}-\\frac{1}{20}=\\frac{109}{120}$$", "input": "\\frac{7}{12}+\\frac{3}{8}-\\frac{1}{20}", "steps": [ { "type": "interim", "title": "Least Common Multiplier of $$12,\\:8,\\:20:{\\quad}120$$", "input": "12,\\:8,\\:20", "steps": [ { "type": "definition", "title": "Least Common Multiplier (LCM)", "text": "The LCM of $$a,\\:b$$ is the smallest positive number that is divisible by both $$a$$ and $$b$$" }, { "type": "interim", "title": "Prime factorization of $$12:{\\quad}2\\cdot\\:2\\cdot\\:3$$", "input": "12", "steps": [ { "type": "step", "primary": "$$12\\:$$divides by $$2\\quad\\:12=6\\cdot\\:2$$", "result": "=2\\cdot\\:6" }, { "type": "step", "primary": "$$6\\:$$divides by $$2\\quad\\:6=3\\cdot\\:2$$", "result": "=2\\cdot\\:2\\cdot\\:3" }, { "type": "step", "primary": "$$2,\\:3$$ are all prime numbers, therefore no further factorization is possible", "result": "=2\\cdot\\:2\\cdot\\:3" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRvTIRluRNPwUULD5JCqpmdzmDmiXg+V79OVxDmBC/OzVB4gitN/2ICkrV6ivfiR3BLFRzd4QlsM8ugKm4vxBIECR4XWPWntAkaT/mGe0ZPFN" } }, { "type": "interim", "title": "Prime factorization of $$8:{\\quad}2\\cdot\\:2\\cdot\\:2$$", "input": "8", "steps": [ { "type": "step", "primary": "$$8\\:$$divides by $$2\\quad\\:8=4\\cdot\\:2$$", "result": "=2\\cdot\\:4" }, { "type": "step", "primary": "$$4\\:$$divides by $$2\\quad\\:4=2\\cdot\\:2$$", "result": "=2\\cdot\\:2\\cdot\\:2" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRtXnBik8vDPXVw+nKWp28DI/y9DKGIPglJ+qMi9xDu2KE1OovxZAaXg7BtrFPk4UcCzRnGgMN6CYRfod7Mq0dp1RcgsS082tQWmOBW6FvhEw" } }, { "type": "interim", "title": "Prime factorization of $$20:{\\quad}2\\cdot\\:2\\cdot\\:5$$", "input": "20", "steps": [ { "type": "step", "primary": "$$20\\:$$divides by $$2\\quad\\:20=10\\cdot\\:2$$", "result": "=2\\cdot\\:10" }, { "type": "step", "primary": "$$10\\:$$divides by $$2\\quad\\:10=5\\cdot\\:2$$", "result": "=2\\cdot\\:2\\cdot\\:5" }, { "type": "step", "primary": "$$2,\\:5$$ are all prime numbers, therefore no further factorization is possible", "result": "=2\\cdot\\:2\\cdot\\:5" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRl8ZboA8wPLg0yhI4RzfjFzwPjJo8Fi+uo29zEltAtWEB4gitN/2ICkrV6ivfiR3BLFRzd4QlsM8ugKm4vxBIEDIX3G4SX/N77p2fbACOqjb" } }, { "type": "step", "primary": "Compute a number comprised of factors that appear in at least one of the following:<br/>$$12,\\:8,\\:20$$", "result": "=2\\cdot\\:2\\cdot\\:2\\cdot\\:3\\cdot\\:5" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2\\cdot\\:2\\cdot\\:3\\cdot\\:5=120$$", "result": "=120" } ], "meta": { "solvingClass": "LCM", "interimType": "LCM Top 1Eq" } }, { "type": "interim", "title": "Adjust Fractions based on the LCM", "steps": [ { "type": "step", "primary": "Multiply each numerator by the same amount needed to multiply its<br/>corresponding denominator to turn it into the LCM $$120$$" }, { "type": "step", "primary": "For $$\\frac{7}{12}:\\:$$multiply the denominator and numerator by $$10$$", "result": "\\frac{7}{12}=\\frac{7\\cdot\\:10}{12\\cdot\\:10}=\\frac{70}{120}" }, { "type": "step", "primary": "For $$\\frac{3}{8}:\\:$$multiply the denominator and numerator by $$15$$", "result": "\\frac{3}{8}=\\frac{3\\cdot\\:15}{8\\cdot\\:15}=\\frac{45}{120}" }, { "type": "step", "primary": "For $$\\frac{1}{20}:\\:$$multiply the denominator and numerator by $$6$$", "result": "\\frac{1}{20}=\\frac{1\\cdot\\:6}{20\\cdot\\:6}=\\frac{6}{120}" } ], "meta": { "interimType": "LCD Adjust Fractions 1Eq" } }, { "type": "step", "result": "=\\frac{70}{120}+\\frac{45}{120}-\\frac{6}{120}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{70+45-6}{120}" }, { "type": "step", "primary": "Add the numbers: $$70+45-6=109$$", "result": "=\\frac{109}{120}" } ], "meta": { "solvingClass": "Solver2" } }, "meta": { "showVerify": true } }