A Father is twice as old as his son. Twelve years ago the father was four times as old as the son was then. Determine their present ages.

{ "query": { "display": "A Father is twice as old as his son. Twelve years ago the father was four times as old as the son was then. Determine their present ages.", "symbolab_question": "#A Father is twice as old as his son. Twelve years ago the father was four times as old as the son was then. Determine their present ages." }, "solution": { "level": "PERFORMED", "subject": "Word Problems", "topic": "Age", "subTopic": "Other", "default": "\\mathrm{\\mathrm{Boy}'s\\:age\\:is:}\\:18<br/>\\mathrm{\\mathrm{Father}'s\\:age\\:is:}\\:36" }, "steps": { "type": "interim", "title": "A Father is twice as old as his boy.<br/>12 years ago the father was 4 times as old as the boy was then.<br/>Determine their present ages<br/> <br/>\\mathrm{Boy}'s age is: $$18$$<br/>\\mathrm{Father}'s age is: $$36$$", "steps": [ { "type": "interim", "title": "Translate the problem into an equation:$${\\quad}2x-12=4\\left(x-12\\right)$$", "steps": [ { "type": "step", "primary": "Represent ages in terms of $$x:$$", "result": "\\mathrm{Boy}\\::\\:x<br/>\\mathrm{Father}\\::\\:2x" }, { "type": "step", "primary": "Ages 12 years ago", "result": "\\mathrm{Boy}\\::\\:x-12<br/>\\mathrm{Father}\\::\\:2x-12" }, { "type": "step", "primary": "Write an equation for their ages 12 years ago", "result": "2x-12=4\\left(x-12\\right)" } ] }, { "type": "interim", "title": "Solve for $$x,\\:2x-12=4\\left(x-12\\right):{\\quad}x=18$$", "input": "2x-12=4\\left(x-12\\right)", "steps": [ { "type": "interim", "title": "Expand $$4\\left(x-12\\right):{\\quad}4x-48$$", "input": "4\\left(x-12\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=4,\\:b=x,\\:c=12$$" ], "result": "=4x-4\\cdot\\:12", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:12=48$$", "result": "=4x-48" } ], "meta": { "solvingClass": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7WwVXNikyUuZVfZ3rwMwCuncUJadsLvGcDY7IUPYXjf6zs903yhxK0NInTwR7JvSRyM/29l7SHlwNleTQoRn0qBJyf8zawtgEaDEKWrMLEzbeQYfUtSvDhv2mev67MlpE" } }, { "type": "step", "result": "2x-12=4x-48" }, { "type": "interim", "title": "Move $$12\\:$$to the right side", "input": "2x-12=4x-48", "result": "2x=4x-36", "steps": [ { "type": "step", "primary": "Add $$12$$ to both sides", "result": "2x-12+12=4x-48+12" }, { "type": "step", "primary": "Simplify", "result": "2x=4x-36" } ], "meta": { "gptData": "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" } }, { "type": "interim", "title": "Move $$4x\\:$$to the left side", "input": "2x=4x-36", "result": "-2x=-36", "steps": [ { "type": "step", "primary": "Subtract $$4x$$ from both sides", "result": "2x-4x=4x-36-4x" }, { "type": "step", "primary": "Simplify", "result": "-2x=-36" } ], "meta": { "gptData": "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" } }, { "type": "interim", "title": "Divide both sides by $$-2$$", "input": "-2x=-36", "result": "x=18", "steps": [ { "type": "step", "primary": "Divide both sides by $$-2$$", "result": "\\frac{-2x}{-2}=\\frac{-36}{-2}" }, { "type": "interim", "title": "Simplify", "input": "\\frac{-2x}{-2}=\\frac{-36}{-2}", "result": "x=18", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{-2x}{-2}:{\\quad}x$$", "input": "\\frac{-2x}{-2}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{-b}=\\frac{a}{b}$$", "result": "=\\frac{2x}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{2}{2}=1$$", "result": "=x" } ], "meta": { "solvingClass": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7KBKbfVC+MMwQLykNKIiEGnyRHuGw7+tM5METTDj6vVEed1oZgJr3Rrt+25B4RF18ZEt3ZXAiqUE0HIXrrrezJHZKl2mV4AlztiApTAFigz2cLDAdDDkGLZBqATtsrhC/ialcV/dI5TH4fXyp+ncwuA==" } }, { "type": "interim", "title": "Simplify $$\\frac{-36}{-2}:{\\quad}18$$", "input": "\\frac{-36}{-2}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{-b}=\\frac{a}{b}$$", "result": "=\\frac{36}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{36}{2}=18$$", "result": "=18" } ], "meta": { "solvingClass": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7HnPzhFzo/XTcUMaxN8qHaHyRHuGw7+tM5METTDj6vVFF4Cz6Dbf5quZuDjS+xHGSP8vQyhiD4JSfqjIvcQ7timkSOxgqdB0M/sw8Nt2sXXS+oSfoJCRRoeYXwJeKgm6cjwE87HTCWyAU3ypRroDMDQ==" } }, { "type": "step", "result": "x=18" } ] } ], "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{Boy}\\::\\:18<br/>\\mathrm{Father}\\::\\:2\\cdot\\:18" }, { "type": "step", "result": "2\\cdot\\:18=36" } ] }, { "type": "step", "result": "\\mathrm{\\mathrm{Boy}'s\\:age\\:is:}\\:18<br/>\\mathrm{\\mathrm{Father}'s\\:age\\:is:}\\:36" } ] } }