[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-143067-105":59,"doc-detail-143067-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","25-challenging-sat-problems-review-practice","25 Challenging SAT Problems - Review & practice","","A set of 25 highly challenging SAT math problems organized by topic and aligned with the progression of the corresponding study book. The problems span Algebra & Functions, Exponentials, Quadratics, Geometry & Trig, Statistics, plus scattered word problems, with many items centered on evaluating multiple options (I, II, III). The collection emphasizes advanced reasoning and deeper conceptual understanding of how SAT math relationships work under pressure.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/25-challenging-sat-problems-review-practice/143067/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/25-challenging-sat-problems-review-practice/143067.png","ImageObject",300,407,{"name":92,"@type":93},"Patrick","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-03","2026-08-25",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",52,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"How are the 25 SAT math problems organized in the set?","Question",{"text":113,"@type":114},"They follow the book’s ordering: Chapter 1 Algebra & Functions (Q1-5), Chapter 2 Exponentials (Q6-8), Chapter 3 Quadratics (Q9-14), scattered word problems (Q15-16), Chapter 4 Geometry & Trig (Q17-22), and Chapter 5 Statistics (Q23-25).","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"What makes these problems particularly difficult?",{"text":118,"@type":114},"The set is based on variations of the hardest SAT problems, and many questions are harder and longer because they cover more concepts in less time, often requiring evaluation of three options (I, II, III).",{"name":120,"@type":111,"acceptedAnswer":121},"How should a student respond if these problems are too challenging?",{"text":122,"@type":114},"The document recommends using the book SAT Math from the Ground Up, which focuses on why the math works rather than only memorizing solution methods for this difficulty level.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},143067,1787692486,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":140,"language":141,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":142,"faqs":143,"seo_title":144,"seo_description":67,"update_tm":130,"read_time":145},549758146520,"https://ap-avatar.wpscdn.com/avatar/80002397d8c0411e94?_k=1775819394049821470","25 Challenging SAT Problems  \n[From learnsatmath.com](From learnsatmath.com)  \nThese questions are all variations of the hardest problems from the Fall 2024 and Spring 2025 SATs. The difficulty of these problems is comparable to the difficulty of the final 3 problems on the hardest module 2 .  \nThis set is also fairly comprehensive, meaning that it covers almost every challenging topic that you can expect to see.  \nIf you find these problems too challenging, checkout my book, SAT Math from the Ground Up at [learnsatmath.com. It will](learnsatmath.com. It will) teach you... you’ll never guess this one... the SAT math from the ground up. Rather than focusing on how to solve problems, it’ll teach you why the math really works, which is essential to tackle problems of this difficulty.  \nThese 25 problems follow the book’s ordering:  \n• Chapter 1: Algebra & Functions-Q1-5  \n• Chapter 2: Exponentials-Q6-8  \n• Chapter 3: Quadratics-Q9-14  \n• Word Problems (scattered): Q15-16  \n• Chapter 4: Geometry & Trig-Q17-22  \n• Chapter 5: Statistics-Q23-25  \nFinal footnote: You may notice that a disproportionate number of these problems have you evaluate 3 options (the I, II, III problems) . This is because:  \n1. These problems are harder/longer  \n2. These problems cover more concepts in less time  \nJust know that on the actual SAT, you won’t be getting nearly as many of these.  \n1 Challenging Functions  \n􀔕(􀔧) = 􀔐 √􀔧 − 􀔑 + 12 √􀔧 − 􀔑  \nFor the function 􀔕, 􀔐 and 􀔑 are constants, and 􀔕(4) = 0 . If 􀔕(5) > 0, which of the following could be a value for 􀔐 + 􀔑?  \nA) -20  \nB) -12  \nC) -8  \nD) -4  \n2 Fraction Algebra 1-Constant Denominators  \n16 16  \n~~ ~~ = ~~ ~~  \n􀔐 􀔑  \n+  \n16  \n􀔒  \nWhich of the following expresses 􀔐 in terms of 􀔑 and 􀔒 ?  \nA) 􀔐 = 􀕽􀕾  \nB) 􀔐 = 􀕽􀕾  \nC) 􀔐 = 􀔑􀔒(􀔑 + 􀔒)  \nD) 􀔐 = 16􀔑􀔒(􀔑 + 􀔒)  \n3 Fraction Algebra 2-Solving Coefficients  \n􀔐􀔧 3 + 􀔑􀔧 2 + 􀔒􀔧 + 􀔓 5 6􀔧  \n~~ ~~ = ~~ ~~ + ~~ ~~  \n(􀔧 2 − 9)(􀔧 + 2) 􀔧 + 3 􀔧 − 3  \nWhat is the value of 􀔐 + 􀔑 + 􀔒?  \n4 Unintuitive Linear Models  \nThere are two lawn mowing companies. Company A charges a flat fee of $80 for the first hour, then an hourly rate of $60 for each additional hour. Company B charges no flat fee, but an hourly rate of $75 per hour. After how many minutes will company A’s total cost exceed company B’s?  \n5 Standard Form Solutions  \n56􀔧 + 􀔐􀔨 = 􀔑 1110􀔧 + 􀔒􀔨 = 2􀔓  \nIf the system of equations above has infinite solutions, what must be the value of 􀕽􀕿?  \n6 Percentages  \n􀔐 is 540% more than 􀔑 and 􀔑 is 0 .02% of 􀔒 .  \nIf 􀔐 is 􀔚% of 􀔒 , then what must 􀔚 equal?  \n7 Exponential Functions 1-Fractional Exponents  \n􀔕(􀔣) = 􀔐(2) 21􀕙1  \nThe given function 􀔕(􀔣) models the amount of money in an investment account 􀔣 years after the account is opened, where 􀔐 is a constant. How many months will it take for the initial investment to increase by 700%?  \n8 Exponential Functions 2-Constants & Coefficients  \n􀔕(􀔧) = 􀔚 3 (􀔚􀖓 ) + 􀔑  \nThe function 􀔕 is defined by the equation above for 􀔧 ≥ 0 . 􀔚 and 􀔑 are positive constantsand 􀔚 \u003C 1 . Which of the following must be true?  \nI. The maximum value of the function is displayed as a constant or coefficient of the function.  \nII. 􀔕(1) = 􀔕(0) ∗ 􀔚  \nIII. 􀔕(􀔐) \u003C 􀔑 for some 􀔐 > 0 .  \nA) I only  \nB) II and III only  \nC) I and III only  \nD) Neither I, II, nor III  \n9 Integer Factors 1-Maximizing Products  \n15􀔧 18 + 􀔚􀔧 9 + 24  \nThe expression above has factors 􀔐􀔧 9 + 􀔑 and 􀔒􀔧 9 + 􀔓, where 􀔐 , 􀔑 , 􀔒 , and 􀔓 are all integer constants. What is the maximum value of 􀔚?  \n10 Integer Factors 2-Identifying Integers  \n(ℎ􀔧 + 􀔚)(􀔠􀔧 + 􀔩) = 􀔐􀔧 2 + 􀔑􀔧 + 35  \nFor the expression above, ℎ, 􀔚 , 􀔠 , and 􀔩 are all integer constants. Which of the following must be an integer?  \nI. 35􀖆  \nII. 􀔑  \nIII. − 􀕼􀖌  \nA) I and II only  \nB) II and III only  \nC) I and III only  \nD) I, II, and III  \n11 Quadratics 1-Vertex Form  \n􀔕(􀔧) = −3􀔧 2 + 􀔑􀔧 + 􀔒  \nIn the given quadratic function, 􀔑 and 􀔒 are constants. The graph of 􀔨 = 􀔕(􀔧) in the xyplane is a parabola that has a vertex at the point (ℎ, 􀔚), where ℎ and 􀔚 are","cbCaiiVOw7myLPge","https://ap.wps.com/l/cbCaiiVOw7myLPge","pdf",74358,15,"English","# Chapter 1: Algebra & Functions\n# Chapter 2: Exponentials\n# Chapter 3: Quadratics\n# Word Problems\n# Chapter 4: Geometry & Trig\n# Chapter 5: Statistics\n# Final note: I, II, III option-evaluation tendency","[{\"question\":\"How are the 25 SAT math problems organized in the set?\",\"answer\":\"They follow the book’s ordering: Chapter 1 Algebra \\u0026 Functions (Q1-5), Chapter 2 Exponentials (Q6-8), Chapter 3 Quadratics (Q9-14), scattered word problems (Q15-16), Chapter 4 Geometry \\u0026 Trig (Q17-22), and Chapter 5 Statistics (Q23-25).\"},{\"question\":\"What makes these problems particularly difficult?\",\"answer\":\"The set is based on variations of the hardest SAT problems, and many questions are harder and longer because they cover more concepts in less time, often requiring evaluation of three options (I, II, III).\"},{\"question\":\"How should a student respond if these problems are too challenging?\",\"answer\":\"The document recommends using the book SAT Math from the Ground Up, which focuses on why the math works rather than only memorizing solution methods for this difficulty level.\"}]","25 Challenging SAT Problems - Review & practice | PDF",38]