Lucy is three times as old as Alex. Lucy is 7 years older than Megan. The sum of their ages is 126

{ "query": { "display": "Lucy is three times as old as Alex. Lucy is 7 years older than Megan. The sum of their ages is 126", "symbolab_question": "#Lucy is three times as old as Alex. Lucy is 7 years older than Megan. The sum of their ages is 126" }, "solution": { "level": "PERFORMED", "subject": "Word Problems", "topic": "Age", "subTopic": "Other", "default": "\\mathrm{\\mathrm{Lucy}'s\\:age\\:is:}\\:57<br/>\\mathrm{\\mathrm{Megan}'s\\:age\\:is:}\\:50<br/>\\mathrm{\\mathrm{Alex}'s\\:age\\:is:}\\:19" }, "steps": { "type": "interim", "title": "Lucy is 3 times as old as Alex.<br/>Lucy is 7 years older than Megan.<br/>The sum of their ages is 126<br/> <br/>\\mathrm{Lucy}'s age is: $$57$$<br/>\\mathrm{Megan}'s age is: $$50$$<br/>\\mathrm{Alex}'s age is: $$19$$", "steps": [ { "type": "interim", "title": "Translate the problem into an equation:$${\\quad}x+\\frac{x}{3}+x-7=126$$", "steps": [ { "type": "step", "primary": "Represent ages in terms of $$x:$$", "result": "\\mathrm{Lucy}\\::\\:x<br/>\\mathrm{Megan}\\::\\:x-7<br/>\\mathrm{Alex}\\::\\:\\frac{x}{3}" }, { "type": "step", "primary": "Write an equation for their ages ", "result": "x+\\frac{x}{3}+x-7=126" } ] }, { "type": "interim", "title": "Solve for $$x,\\:x+\\frac{x}{3}+x-7=126:{\\quad}x=57$$", "input": "x+\\frac{x}{3}+x-7=126", "steps": [ { "type": "step", "primary": "Add similar elements: $$x+x=2x$$", "result": "2x+\\frac{x}{3}-7=126" }, { "type": "interim", "title": "Move $$7\\:$$to the right side", "input": "2x+\\frac{x}{3}-7=126", "result": "2x+\\frac{x}{3}=133", "steps": [ { "type": "step", "primary": "Add $$7$$ to both sides", "result": "2x+\\frac{x}{3}-7+7=126+7" }, { "type": "step", "primary": "Simplify", "result": "2x+\\frac{x}{3}=133" } ], "meta": { "gptData": "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" } }, { "type": "interim", "title": "Multiply both sides by $$3$$", "input": "2x+\\frac{x}{3}=133", "result": "7x=399", "steps": [ { "type": "step", "primary": "Multiply both sides by $$3$$", "result": "2x\\cdot\\:3+\\frac{x}{3}\\cdot\\:3=133\\cdot\\:3" }, { "type": "interim", "title": "Simplify", "input": "2x\\cdot\\:3+\\frac{x}{3}\\cdot\\:3=133\\cdot\\:3", "result": "7x=399", "steps": [ { "type": "interim", "title": "Simplify $$2x\\cdot\\:3:{\\quad}6x$$", "input": "2x\\cdot\\:3", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:3=6$$", "result": "=6x" } ], "meta": { "solvingClass": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7UXsWYAhnfEQNQXKj7hLmqyAn9lkDfZkicUGkO3EF+IokbdOjr06uz6UehgahgPghgQUxJPyUNnGfVirkcwpVO3RNt9xNvrn0uR65TGUcKLrBDbnNZ0x55I4zIqkoxd+U" } }, { "type": "interim", "title": "Simplify $$\\frac{x}{3}\\cdot\\:3:{\\quad}x$$", "input": "\\frac{x}{3}\\cdot\\:3", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{x\\cdot\\:3}{3}" }, { "type": "step", "primary": "Cancel the common factor: $$3$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79/ON7FFMhjzCCH2STVspJCPpFLppSOOOwknMsNOrv36rju+5Z51e/ZZSD3gRHwjBvZTuV3pU1o5URU/sdf9xYZjcBIL5pmo83UMFZRSzJsWQA7yHY4blmznYtOPht2hnnQx8T++oEXHxO9/c1XQElQ==" } }, { "type": "interim", "title": "Simplify $$133\\cdot\\:3:{\\quad}399$$", "input": "133\\cdot\\:3", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$133\\cdot\\:3=399$$", "result": "=399" } ], "meta": { "solvingClass": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7iAp4Wz783xKkna14oAylA96GQqufR6tr2vPxOUv7H+/16VAIqkowbC0SDlJfY8ecRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6qM9tWw+zE8gVSHp66gmw7o" } }, { "type": "step", "result": "6x+x=399" }, { "type": "step", "result": "7x=399" } ] } ], "meta": { "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$7$$", "input": "7x=399", "result": "x=57", "steps": [ { "type": "step", "primary": "Divide both sides by $$7$$", "result": "\\frac{7x}{7}=\\frac{399}{7}" }, { "type": "step", "primary": "Simplify", "result": "x=57" } ], "meta": { "gptData": "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" } } ], "meta": { "solvingClass": "Equations" } }, { "type": "interim", "title": "Find age", "steps": [ { "type": "step", "primary": "Plug in the value for $$x\\:$$to find current ages:", "result": "\\mathrm{Lucy}\\::\\:57<br/>\\mathrm{Megan}\\::\\:57-7<br/>\\mathrm{Alex}\\::\\:\\frac{57}{3}" }, { "type": "interim", "title": "Convert improper fractions to mixed numbers:$${\\quad}\\frac{57}{3}=19$$", "steps": [ { "type": "interim", "title": "$$\\frac{57}{3}=19\\quad\\:$$Remainder$$\\quad\\:0$$", "input": "\\frac{57}{3}", "steps": [ { "type": "step", "primary": "Write the problem in long division format", "result": "\\begin{matrix}3\\overline{|\\kern0.1em57}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\end{matrix}" }, { "type": "interim", "title": "Divide $$5\\:$$by $$3\\:$$to get $$1$$", "input": "\\:", "result": "\\begin{array}{l}\\:\\:\\:\\:1\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\3\\overline{|\\kern0.1em57}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\\\:\\:\\:\\:\\underline{3}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\\\:\\:\\:\\:27\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\end{array}", "steps": [ { "type": "step", "primary": "Divide $$5\\:$$by $$3\\:$$to get $$1$$", "result": "\\begin{array}{l}\\:\\:\\:\\:1\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\3\\overline{|\\kern0.1em57}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\end{array}" }, { "type": "step", "primary": "Multiply the quotient digit $$\\left(1\\right)\\:$$by the divisor $$3$$", "result": "\\begin{array}{l}\\:\\:\\:\\:1\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\3\\overline{|\\kern0.1em57}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\\\:\\:\\:\\:\\underline{3}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\end{array}" }, { "type": "step", "primary": "Subtract $$3\\:$$from $$5$$", "result": "\\begin{array}{l}\\:\\:\\:\\:1\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\3\\overline{|\\kern0.1em57}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\\\:\\:\\:\\:\\underline{3}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\\\:\\:\\:\\:2\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\end{array}" }, { "type": "step", "primary": "Bring down the next number of the dividend", "result": "\\begin{array}{l}\\:\\:\\:\\:1\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\3\\overline{|\\kern0.1em57}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\\\:\\:\\:\\:\\underline{3}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\\\:\\:\\:\\:27\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\end{array}" } ], "meta": { "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjAuRbUqF0v2PpVuPGQH4dLbLeu+MH3/4O31zRKyMpNuXtAM4L+5fXXw9GSOcy6r171uNUoGmF+soHjoTRCzmAYKuxG4H5MMdb7eJWj/ZjJHObnxrODmx8dzZSf6H8RiCJYoO0Q00Z8xMW8/k3rg4tLA72YfaApE1ycD6mkgxf1ySIBJVxB5VOQ3NcZmXv/ESPY=" } }, { "type": "interim", "title": "Divide $$27\\:$$by $$3\\:$$to get $$9$$", "input": "\\:", "result": "\\begin{array}{l}\\:\\:\\:\\:19\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\3\\overline{|\\kern0.1em57}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\\\:\\:\\:\\:\\underline{3}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\\\:\\:\\:\\:27\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\\\:\\:\\:\\:\\underline{27}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\\\:\\:\\:\\:\\:\\:0\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\end{array}", "steps": [ { "type": "step", "primary": "Divide $$27\\:$$by $$3\\:$$to get $$9$$", "result": "\\begin{array}{l}\\:\\:\\:\\:19\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\3\\overline{|\\kern0.1em57}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\\\:\\:\\:\\:\\underline{3}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\\\:\\:\\:\\:27\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\end{array}" }, { "type": "step", "primary": "Multiply the quotient digit $$\\left(9\\right)\\:$$by the divisor $$3$$", "result": "\\begin{array}{l}\\:\\:\\:\\:19\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\3\\overline{|\\kern0.1em57}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\\\:\\:\\:\\:\\underline{3}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\\\:\\:\\:\\:27\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\\\:\\:\\:\\:\\underline{27}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\end{array}" }, { "type": "step", "primary": "Subtract $$27\\:$$from $$27$$", "result": "\\begin{array}{l}\\:\\:\\:\\:19\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\3\\overline{|\\kern0.1em57}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\\\:\\:\\:\\:\\underline{3}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\\\:\\:\\:\\:27\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\\\:\\:\\:\\:\\underline{27}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\\\:\\:\\:\\:\\:\\:0\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\end{array}" } ], "meta": { "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjAuRbUqF0v2PpVuPGQH4dLbLeu+MH3/4O31zRKyMpNuXtAM4L+5fXXw9GSOcy6r170n8Q6DU4dR1Ul+r2ffZ9qYG3RwD/BlFf6PiNRpfzRS75EQlrQ67quu86RDujFLtS5oXGeCRJ/u5pJw39XdfvghCov3X8Rmv87zdT716PEIoFlkR9J0/g3Ruo5oBU8EgCs=" } }, { "type": "step", "primary": "The solution for Long Division of $$\\frac{57}{3}\\:$$is $$19\\:$$with remainder of $$0$$", "result": "19\\quad\\:\\mathrm{Remainder}\\quad\\:0" } ], "meta": { "solvingClass": "StepsLongDivisionDecimals" } }, { "type": "step", "primary": "Convert to mixed number: Quotient$$\\frac{\\mathrm{Remainder}}{\\mathrm{Divisor}}$$", "secondary": [ "$$\\frac{57}{3}=19$$" ] }, { "type": "step", "result": "=19" } ] }, { "type": "step", "result": "57-7=50" } ] }, { "type": "step", "result": "\\mathrm{\\mathrm{Lucy}'s\\:age\\:is:}\\:57<br/>\\mathrm{\\mathrm{Megan}'s\\:age\\:is:}\\:50<br/>\\mathrm{\\mathrm{Alex}'s\\:age\\:is:}\\:19" } ] } }