[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-167354-en":3,"doc-seo-167354-105":30,"detail-sidebar-cat-0-en-105":90},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},167354,1374391974585,"Genevieve","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",4,"Exam","Gears & Work-Rate Problems - A Complete Study Guidebook","A complete study guidebook on work-rate problems and gear systems, expanding handwritten notes with clear explanations, diagrams, and 20 leveled practice questions. Work-based sections introduce key relationships like V = p × h and combining worker rates using 1/x1 + 1/x2 = 1/x0, reinforced through worked examples and timing insights. Gear sections cover RPM, teeth count, mesh behavior, the gear law (RPM_A · T_A = RPM_B · T_B), gear ratio interpretation, and rotation direction rules with multi-gear chains and idlers.","Gears & Work-Rate Problems\nA complete study guidebook\nBased on handwritten copybook notes, expanded with explanations, diagrams, and 20 leveled practice questions.\n\u000f\nTable of Contents\n\u0003\u0013TOC \\h \\o \"1-3\"\u0014\n\u0015\u0004\n\u000f\nPart 1 — Work-Based Problems\nThese problems model situations where a job gets done at a certain rate — people working over hours, taps filling a tank, machines producing units, etc.\nKey formula\nVolume (work done) = rate × time  →  V = p × h\nWhere p = people (or units of work-rate) and h = hours. This is the general \"work = rate × time\" idea.\nCombining two workers\nIf two people can each finish a job alone in different amounts of time, their combined rate is the sum of their individual rates:\n1/x₁ + 1/x₂ = 1/x₀\nwhere x₁ and x₂ are the times each person needs alone, and x₀ is the time they need working together.\nWorked example from the notes\nTom can finish a job alone in 3 hours. Jack can finish the same job alone in 7 hours. How long do they take working together?\n1/3 + 1/7 = 1/x₀\nFinding a common denominator (21): 7/21 + 3/21 = 10/21 = 1/x₀\nSo x₀ = 21/10 = 2.1 hours — working together, Tom and Jack finish the job in 2.1 hours (2 hours 6 minutes).\nTip: the combined time is always shorter than either person's individual time.\nTip: this same 1/x₁ + 1/x₂ = 1/x₀ pattern also applies to pipes filling a tank, or any two rates combining.\n\u000f\nPart 2 — Gears\nGears are toothed wheels that transmit rotational motion. When two gears mesh, their teeth interlock so that the speed of teeth passing the contact point is identical for both gears.\nKey vocabulary\nRPM — rotations per minute (how fast a gear spins)\nT — number of teeth on a gear\nMesh — when two gears' teeth interlock directly\nIdler gear — a middle gear in a chain that changes direction but does not affect the overall speed ratio between the first and last gear\nThe fundamental gear law\nRPM_A · T_A = RPM_B · T_B\nThis says: teeth passing the mesh point per minute is the same for both gears. A gear with fewer teeth must spin faster to keep up with a gear that has more teeth.\nRule of thumb: fewer teeth → faster spin. More teeth → slower, stronger spin (more torque).\nGear ratio\nGear ratio (G.R.) = T_B / T_A = RPM_A / RPM_B\nThe gear ratio tells you two things at once: how many times bigger gear B is (in teeth) compared to A, and how many times faster A spins compared to B.\nWorked example\nIf T_B = 60 and T_A = 20, then G.R. = 60/20 = 3. Gear B has 3× the teeth of A, so gear B spins 3× slower than gear A.\nDirection of rotation\nThe direction rule depends on HOW the gears are connected:\nGear chains — the even/odd rule\nFor a row of gears each meshing directly with the next (A → B → C → D...), each mesh flips the direction. So:\nEven number of gears in the chain → the last gear spins OPPOSITE to the first\nOdd number of gears in the chain → the last gear spins the SAME direction as the first\nIdler gears and the overall ratio\nIn a chain of gears, only the FIRST and LAST gear's teeth counts matter for the overall speed ratio — middle gears (idlers) change direction and fill space but their own teeth count cancels out of the final ratio.\nRPM_last / RPM_first = T_first / T_last\n\u000f\nPart 3 — Practice Questions (Easy → Expert)\n20 questions covering everything above. Try each one before reading the explanation.\nQuestion 1  [Easy]\nAnswer & explanation\nUsing RPM_A · T_A = RPM_B · T_B:  90 × 30 = RPM_B × 10  →  RPM_B = 2700/10 = 270 RPM.\nGear B has 1/3 the teeth of A, so it spins 3× faster.\nQuestion 2  [Easy]\nAnswer & explanation\nGears on the same shaft always share the same RPM and direction. Y's RPM = 200 RPM (the teeth counts do not matter here).\nQuestion 3  [Easy]\nAnswer & explanation\nA regular (uncrossed) belt keeps both pulleys turning the SAME direction. Gear B spins clockwise.\nQuestion 4  [Easy]\nAnswer & explanation\nGear ratio = 36/12 = 3. Gear A has fewer teeth, so gear A spins faster (3× faster than B).\nQuestion 5  [Easy]\nAnswer & explanation\nMesh A→B flips direction: B spins counter-clockwise. Cros","cbCaiuvWjvFwZeGG","https://ap.wps.com/l/cbCaiuvWjvFwZeGG","docx",293515,1,11,"English","en",105,"# Part 1 — Work-Based Problems\n## Key formula\n## Combining two workers\n## Worked example from the notes\n# Part 2 — Gears\n## Key vocabulary\n## The fundamental gear law\n## Gear ratio\n## Direction of rotation\n## Worked example\n# Part 3 — Practice Questions (Easy → Expert)\n## Question 1–20","[{\"question\":\"What is the core formula for work-rate problems in this guidebook?\",\"answer\":\"It uses Volume (work done) = rate × time, written as V = p × h, where p is people (or work-rate) and h is hours.\"},{\"question\":\"How do you combine two workers when each takes a different time alone?\",\"answer\":\"Their combined rate is found with 1/x1 + 1/x2 = 1/x0, where x1 and x2 are individual times and x0 is the working-together time.\"},{\"question\":\"How is gear speed related to the number of teeth?\",\"answer\":\"The fundamental gear law is RPM_A · T_A = RPM_B · T_B, meaning the gear with fewer teeth must spin faster to match the same teeth passing the mesh point per minute.\"}]","Gears & Work-Rate Problems - A Complete Study Guidebook | DOCX",1788212062,28,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":85,"head_meta":87,"extra_data":89,"updated_unix":28},"gears-work-rate-problems-a-complete-study-guidebook","",{"@graph":36,"@context":84},[37,53,67],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/exam/",3,{"item":52,"name":13,"@type":43,"position":11},"https://docshare.wps.com/document/gears-work-rate-problems-a-complete-study-guidebook/167354/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/vnd.openxmlformats-officedocument.wordprocessingml.document","2026-08-31",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What is the core formula for work-rate problems in this guidebook?","Question",{"text":74,"@type":75},"It uses Volume (work done) = rate × time, written as V = p × h, where p is people (or work-rate) and h is hours.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How do you combine two workers when each takes a different time alone?",{"text":79,"@type":75},"Their combined rate is found with 1/x1 + 1/x2 = 1/x0, where x1 and x2 are individual times and x0 is the working-together time.",{"name":81,"@type":72,"acceptedAnswer":82},"How is gear speed related to the number of teeth?",{"text":83,"@type":75},"The fundamental gear law is RPM_A · T_A = RPM_B · T_B, meaning the gear with fewer teeth must spin faster to match the same teeth passing the mesh point per minute.","https://schema.org",{"og:url":52,"og:type":86,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":88,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":91},[92,96,100,103,108,113,118,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":93,"show_sort_weight":94,"slug":95},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":97,"show_sort_weight":98,"slug":99},"Literature",80,"literature",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":101,"slug":102},70,"exam",{"id":104,"doc_module":4,"doc_module_name":46,"category_name":105,"show_sort_weight":106,"slug":107},5,"Comic",60,"comic",{"id":109,"doc_module":4,"doc_module_name":46,"category_name":110,"show_sort_weight":111,"slug":112},6,"Technology",50,"technology",{"id":114,"doc_module":4,"doc_module_name":46,"category_name":115,"show_sort_weight":116,"slug":117},7,"Healthcare",40,"healthcare",{"id":119,"doc_module":4,"doc_module_name":46,"category_name":120,"show_sort_weight":121,"slug":122},8,"Research & Report",30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":104,"slug":138},19,"General","general"]