Lauren's age is half of Joe's age. Emma is four years older than Joe. The sum of Lauren, Emma, and Joe's age is 80. How old is Joe?

{ "query": { "display": "Lauren's age is half of Joe's age. Emma is four years older than Joe. The sum of Lauren, Emma, and Joe's age is 80. How old is Joe?", "symbolab_question": "#Lauren's age is half of Joe's age. Emma is four years older than Joe. The sum of Lauren, Emma, and Joe's age is 80. How old is Joe?" }, "solution": { "level": "PERFORMED", "subject": "Word Problems", "topic": "Age", "subTopic": "Other", "default": "\\mathrm{\\mathrm{Joe}'s\\:age\\:is:}\\:30\\frac{2}{5}" }, "steps": { "type": "interim", "title": "Lauren's age is 1/2 of Joe's age.<br/>Emma is 4 years older than Joe.<br/>The sum of Lauren Emma and Joe's age is 80.<br/>How old is Joe?<br/> <br/>\\mathrm{Joe}'s age is: $$30\\frac{2}{5}$$", "steps": [ { "type": "interim", "title": "Translate the problem into an equation:$${\\quad}\\frac{1}{2}x+x+4+x=80$$", "steps": [ { "type": "step", "primary": "Represent ages in terms of $$x:$$", "result": "\\mathrm{Joe}\\::\\:x<br/>\\mathrm{Emma}\\::\\:x+4<br/>\\mathrm{Lauren}\\::\\:\\frac{1}{2}x" }, { "type": "step", "primary": "Write an equation for their ages ", "result": "\\frac{1}{2}x+x+4+x=80" } ] }, { "type": "interim", "title": "Solve for $$x,\\:\\frac{1}{2}x+x+4+x=80:{\\quad}x=\\frac{152}{5}$$", "input": "\\frac{1}{2}x+x+4+x=80", "steps": [ { "type": "step", "primary": "Group like terms", "result": "\\frac{1}{2}x+x+x+4=80" }, { "type": "step", "primary": "Add similar elements: $$x+x=2x$$", "result": "\\frac{1}{2}x+2x+4=80" }, { "type": "interim", "title": "Move $$4\\:$$to the right side", "input": "\\frac{1}{2}x+2x+4=80", "result": "\\frac{1}{2}x+2x=76", "steps": [ { "type": "step", "primary": "Subtract $$4$$ from both sides", "result": "\\frac{1}{2}x+2x+4-4=80-4" }, { "type": "step", "primary": "Simplify", "result": "\\frac{1}{2}x+2x=76" } ], "meta": { "gptData": "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" } }, { "type": "interim", "title": "Multiply both sides by $$2$$", "input": "\\frac{1}{2}x+2x=76", "result": "5x=152", "steps": [ { "type": "step", "primary": "Multiply both sides by $$2$$", "result": "\\frac{1}{2}x\\cdot\\:2+2x\\cdot\\:2=76\\cdot\\:2" }, { "type": "interim", "title": "Simplify", "input": "\\frac{1}{2}x\\cdot\\:2+2x\\cdot\\:2=76\\cdot\\:2", "result": "5x=152", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{1}{2}x\\cdot\\:2:{\\quad}x$$", "input": "\\frac{1}{2}x\\cdot\\:2", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{1\\cdot\\:2}{2}x" }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=x\\cdot\\:1" }, { "type": "step", "primary": "Multiply: $$x\\cdot\\:1=x$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7TayuIlGCNKzKncs66z+Jzv4CsFtYCGVpnwQ6GR+/iwQJQJZuTAY5js+oqjdT8ksls9TeFoc2NYtMR3HLuoQQfosaY2cs9YZ/08feWmaOgkhY2wgLqGp4cI6P/WzKf+9pwhztzjvVYzj7cLQmsrJ9hg==" } }, { "type": "interim", "title": "Simplify $$2x\\cdot\\:2:{\\quad}4x$$", "input": "2x\\cdot\\:2", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=4x" } ], "meta": { "solvingClass": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7bb+glc6253U2NNlq0+VfJCAn9lkDfZkicUGkO3EF+IqEnXlKdJOvXJDOfm408RPrgQUxJPyUNnGfVirkcwpVO3RNt9xNvrn0uR65TGUcKLqyGKuJM4vlS6f3PxoaigvX" } }, { "type": "interim", "title": "Simplify $$76\\cdot\\:2:{\\quad}152$$", "input": "76\\cdot\\:2", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$76\\cdot\\:2=152$$", "result": "=152" } ], "meta": { "solvingClass": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7QN9EH3E1tpLbQiuwdoNMOSAn9lkDfZkicUGkO3EF+IofBOTd0IbSzFGMBAfQjjQI72wZm7kDUxdE6YSmfEbr2rjwYWrTTdgwnKiNNe98H1SPQXSVt/bAVZgdLxhGHAQh" } }, { "type": "step", "result": "x+4x=152" }, { "type": "step", "result": "5x=152" } ] } ], "meta": { "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYk4zfdAH4s/wgqI5AuJ55/i95EI2JyU+2RT0qArNxlCqY0KINx7aCCcjHY7WaAbVDOgoRPK/2Tf/pGvcKW+heVZ2AG77HmWXerTWUbxCwIO7soh1N1bMWdCb5YM9Q4f9ltMyKXqjig1oe8nVG8YCoXV694Ph869lyDoQUdss535rzBznFLEvoeX7WBJxz9xoUfwLFXEBow6f3P2eMiFvIcIWF5mVTswXJsTuHrXOCLb54iDZDnxHfIi5ekb8IKx2QeZXlgM6+IwVNxpobAIQsDfzW2ACWhp+t5qlz/1lyi1oyiiirxkwNm01Vu+GwKQ0rF4xSl1IT79oyrZOJpW8Ds33Rb94SEZ/ldjWP3IfHIpylnenT7OF2d9W75/QEKjqVzfiguftzSeHNczvBup8dIhZ9VXfbSWYapymUWw0lt8JNJFpieUqwSM10mklZi3jyKaaZ8ef2sav2E0ahdMP64rO6Zx1D8371h76W0d6Z98dGdF4kg2/TXlR60HycP62paELNNPsfYsgV6SoqTQoVg5jQog3HtoIJyMdjtZoBtUMacgOEU+0nkn/wJwO6nfIJQ==" } }, { "type": "interim", "title": "Divide both sides by $$5$$", "input": "5x=152", "result": "x=\\frac{152}{5}", "steps": [ { "type": "step", "primary": "Divide both sides by $$5$$", "result": "\\frac{5x}{5}=\\frac{152}{5}" }, { "type": "step", "primary": "Simplify", "result": "x=\\frac{152}{5}" } ], "meta": { "gptData": "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" } } ], "meta": { "solvingClass": "Equations" } }, { "type": "interim", "title": "Convert improper fractions to mixed numbers:$${\\quad}30\\frac{2}{5}$$", "steps": [ { "type": "interim", "title": "$$\\frac{152}{5}=30\\quad\\:$$Remainder$$\\quad\\:2$$", "input": "\\frac{152}{5}", "steps": [ { "type": "step", "primary": "Write the problem in long division format", "result": "\\begin{matrix}5\\overline{|\\kern0.1em152}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\end{matrix}" }, { "type": "interim", "title": "Divide $$15\\:$$by $$5\\:$$to get $$3$$", "input": "\\:", "result": "\\begin{array}{l}\\:\\:\\:\\:\\:\\:3\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\5\\overline{|\\kern0.1em152}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\\\:\\:\\:\\:\\underline{15}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\\\:\\:\\:\\:\\:\\:02\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\end{array}", "steps": [ { "type": "step", "primary": "Divide $$15\\:$$by $$5\\:$$to get $$3$$", "result": "\\begin{array}{l}\\:\\:\\:\\:\\:\\:3\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\5\\overline{|\\kern0.1em152}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\end{array}" }, { "type": "step", "primary": "Multiply the quotient digit $$\\left(3\\right)\\:$$by the divisor $$5$$", "result": "\\begin{array}{l}\\:\\:\\:\\:\\:\\:3\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\5\\overline{|\\kern0.1em152}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\\\:\\:\\:\\:\\underline{15}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\end{array}" }, { "type": "step", "primary": "Subtract $$15\\:$$from $$15$$", "result": "\\begin{array}{l}\\:\\:\\:\\:\\:\\:3\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\5\\overline{|\\kern0.1em152}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\\\:\\:\\:\\:\\underline{15}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\\\:\\:\\:\\:\\:\\:0\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\end{array}" }, { "type": "step", "primary": "Bring down the next number of the dividend", "result": "\\begin{array}{l}\\:\\:\\:\\:\\:\\:3\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\5\\overline{|\\kern0.1em152}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\\\:\\:\\:\\:\\underline{15}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\\\:\\:\\:\\:\\:\\:02\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\end{array}" } ], "meta": { "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjAuRbUqF0v2PpVuPGQH4dLbLeu+MH3/4O31zRKyMpNuXtAM4L+5fXXw9GSOcy6r171yGZTzNDLFeFwejKTq04yGN76yOdQE+UYkuJOoiA4vj5EQlrQ67quu86RDujFLtS4efJdNwMEp70pmt1YrRQ0smAN6jM2iQD4FjCSqUHgsiFg7AsTWz/aj7wW4rktN5/U=" } }, { "type": "interim", "title": "Divide $$2\\:$$by $$5\\:$$to get $$0$$", "input": "\\:", "result": "\\begin{array}{l}\\:\\:\\:\\:\\:\\:30\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\5\\overline{|\\kern0.1em152}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\\\:\\:\\:\\:\\underline{15}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\\\:\\:\\:\\:\\:\\:02\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\\\:\\:\\:\\:\\:\\:\\:\\:\\underline{0}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\\\:\\:\\:\\:\\:\\:\\:\\:2\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\end{array}", "steps": [ { "type": "step", "primary": "Divide $$2\\:$$by $$5\\:$$to get $$0$$", "result": "\\begin{array}{l}\\:\\:\\:\\:\\:\\:30\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\5\\overline{|\\kern0.1em152}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\\\:\\:\\:\\:\\underline{15}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\\\:\\:\\:\\:\\:\\:02\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\end{array}" }, { "type": "step", "primary": "Multiply the quotient digit $$\\left(0\\right)\\:$$by the divisor $$5$$", "result": "\\begin{array}{l}\\:\\:\\:\\:\\:\\:30\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\5\\overline{|\\kern0.1em152}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\\\:\\:\\:\\:\\underline{15}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\\\:\\:\\:\\:\\:\\:02\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\\\:\\:\\:\\:\\:\\:\\:\\:\\underline{0}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\end{array}" }, { "type": "step", "primary": "Subtract $$0\\:$$from $$2$$", "result": "\\begin{array}{l}\\:\\:\\:\\:\\:\\:30\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\5\\overline{|\\kern0.1em152}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\\\:\\:\\:\\:\\underline{15}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\\\:\\:\\:\\:\\:\\:02\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\\\:\\:\\:\\:\\:\\:\\:\\:\\underline{0}\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\\\\\:\\:\\:\\:\\:\\:\\:\\:2\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\end{array}" } ], "meta": { "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjAuRbUqF0v2PpVuPGQH4dLbLeu+MH3/4O31zRKyMpNuXtAM4L+5fXXw9GSOcy6r172XmCsY0dJtZGPCmr1vH47xzP7ovb9THepx85Rpn8irs7nxrODmx8dzZSf6H8RiCJa7BjVmIcI6ZeQgl8237pODo7lPw9N3G2C1HSvsokLblI4V7GaPzKCvkEVvnnf/0yo=" } }, { "type": "step", "primary": "The solution for Long Division of $$\\frac{152}{5}\\:$$is $$30\\:$$with remainder of $$2$$", "result": "30\\quad\\:\\mathrm{Remainder}\\quad\\:2" } ], "meta": { "solvingClass": "StepsLongDivisionDecimals" } }, { "type": "step", "primary": "Convert to mixed number: Quotient$$\\frac{\\mathrm{Remainder}}{\\mathrm{Divisor}}$$", "secondary": [ "$$\\frac{152}{5}=30\\frac{2}{5}$$" ] }, { "type": "step", "result": "30\\frac{2}{5}" } ] }, { "type": "step", "result": "\\mathrm{\\mathrm{Joe}'s\\:age\\:is:}\\:30\\frac{2}{5}" } ] } }