Bob's age is twice that of Barry's. Six years ago, Bob was three times older than Barry. Find the age of both.

{ "query": { "display": "Bob's age is twice that of Barry's. Six years ago, Bob was three times older than Barry. Find the age of both.", "symbolab_question": "#Bob's age is twice that of Barry's. Six years ago, Bob was three times older than Barry. Find the age of both." }, "solution": { "level": "PERFORMED", "subject": "Word Problems", "topic": "Age", "subTopic": "Other", "default": "\\mathrm{\\mathrm{Barry}'s\\:age\\:is:}\\:12<br/>\\mathrm{\\mathrm{Bob}'s\\:age\\:is:}\\:24" }, "steps": { "type": "interim", "title": "Bob's age is twice that of Barry's.<br/>6 years ago Bob was 3 times older than Barry.<br/>Find the age of both<br/> <br/>\\mathrm{Barry}'s age is: $$12$$<br/>\\mathrm{Bob}'s age is: $$24$$", "steps": [ { "type": "interim", "title": "Translate the problem into an equation:$${\\quad}2x-6=3\\left(x-6\\right)$$", "steps": [ { "type": "step", "primary": "Represent ages in terms of $$x:$$", "result": "\\mathrm{Barry}\\::\\:x<br/>\\mathrm{Bob}\\::\\:2x" }, { "type": "step", "primary": "Ages 6 years ago", "result": "\\mathrm{Barry}\\::\\:x-6<br/>\\mathrm{Bob}\\::\\:2x-6" }, { "type": "step", "primary": "Write an equation for their ages 6 years ago", "result": "2x-6=3\\left(x-6\\right)" } ] }, { "type": "interim", "title": "Solve for $$x,\\:2x-6=3\\left(x-6\\right):{\\quad}x=12$$", "input": "2x-6=3\\left(x-6\\right)", "steps": [ { "type": "interim", "title": "Expand $$3\\left(x-6\\right):{\\quad}3x-18$$", "input": "3\\left(x-6\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=3,\\:b=x,\\:c=6$$" ], "result": "=3x-3\\cdot\\:6", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:6=18$$", "result": "=3x-18" } ], "meta": { "solvingClass": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+7A0Dz3STE7MoRzjqYVOAQsBmUOGgB0VAXoAXik9qC9EoApuujwRq/Ms3wa4gNT89ZVSO0B13oSpK+zJfqv7kB7MWKURN+43KCOzRc+RXaEARgwgTJxShnqxwbjuZpp/" } }, { "type": "step", "result": "2x-6=3x-18" }, { "type": "interim", "title": "Move $$6\\:$$to the right side", "input": "2x-6=3x-18", "result": "2x=3x-12", "steps": [ { "type": "step", "primary": "Add $$6$$ to both sides", "result": "2x-6+6=3x-18+6" }, { "type": "step", "primary": "Simplify", "result": "2x=3x-12" } ], "meta": { "gptData": "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" } }, { "type": "interim", "title": "Move $$3x\\:$$to the left side", "input": "2x=3x-12", "result": "-x=-12", "steps": [ { "type": "step", "primary": "Subtract $$3x$$ from both sides", "result": "2x-3x=3x-12-3x" }, { "type": "step", "primary": "Simplify", "result": "-x=-12" } ], "meta": { "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$-1$$", "input": "-x=-12", "result": "x=12", "steps": [ { "type": "step", "primary": "Divide both sides by $$-1$$", "result": "\\frac{-x}{-1}=\\frac{-12}{-1}" }, { "type": "step", "primary": "Simplify", "result": "x=12" } ], "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{Barry}\\::\\:12<br/>\\mathrm{Bob}\\::\\:2\\cdot\\:12" }, { "type": "step", "result": "2\\cdot\\:12=24" } ] }, { "type": "step", "result": "\\mathrm{\\mathrm{Barry}'s\\:age\\:is:}\\:12<br/>\\mathrm{\\mathrm{Bob}'s\\:age\\:is:}\\:24" } ] } }