[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-266385-105":59,"doc-detail-266385-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","sat-math-hard-practice-quiz-numbers-and-operations","SAT Math Hard Practice Quiz - Numbers and Operations","","Challenging SAT-style multiple-choice questions covering number theory, algebraic reasoning, and functional relationships. Problems explore ratios and removal changes, monotonic digit integers, arithmetic sequences, factor counting for powers of primes, number properties using divisibility and parity, perfect-square constraints, and consecutive even-integer summations. Additional items require digit rearrangement optimization and analyzing expressions from subtraction and factor patterns. Suitable for focused practice and skill reinforcement.",{"@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/sat-math-hard-practice-quiz-numbers-and-operations/266385/",{"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/sat-math-hard-practice-quiz-numbers-and-operations/266385.png","ImageObject",300,407,{"name":92,"@type":93},"Aran","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-18","2026-09-14",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",16,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What types of number theory concepts appear in the Numbers and Operations section?","Question",{"text":113,"@type":114},"The section includes ratios with removals, monotonic digit integers, prime power factor counting, divisibility by two primes, perfect-square constraints, and relationships among consecutive even integers.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How do the sequence and rearrangement problems test SAT skills?",{"text":118,"@type":114},"The sequence question uses a defined starting term and a constant step to compute a specific term, while the rearrangement question maximizes the difference between two numbers formed from the same nonzero digits.",{"name":120,"@type":111,"acceptedAnswer":121},"What kinds of algebra and functions topics are covered after Numbers and Operations?",{"text":122,"@type":114},"The Algebra and Functions section includes inequality solving for integer parameters, equivalent-expression transformations, simplifying radicals involving exponents, linear range reasoning, solving for zeros via product equations, and function equality/graph symmetry questions.","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},266385,1789406971,{"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},137455076865,"https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0","Numbers and Operations  \n1. A bag contains tomatoes that are either green or red. The ratio of green tomatoes to red tomatoes in the bag is 4 to 3 . When 􀀌ve green tomatoes and 􀀌ve red tomatoes are removed, the ratio becomes 3 to 2 . How many red tomatoes were originally in the bag?  \n(A) 12  \n(B) 15  \n(C) 18  \n(D) 24  \n(E) 30  \n2. If each digit in an integer is greater than the digit to the left, the integer is said to be “monotonic”. For example, 12 is a monotonic integer since 2 > 1. How many positive two-digit monotonic integers are there?  \n(A) 28  \n(B) 32  \n(C) 36  \n(D) 40  \n(E) 44  \na, 2a − 1, 3a − 2, 4a − 3, ...  \n3. For a particular number a, the 􀀌rst term in the sequence above is equal to a, and each term thereafter is 7 greater than the previous term. What is the value of the 16th term in the sequence?  \n4. If p is a prime number, how many factors does p3 have?  \n(A) One  \n(B) Two  \n(C) Three  \n(D) Four  \n(E) Five  \n5. How many integers between 10 and 500 begin and end in 3 ?  \n6. A particular integer N is divisible by two di􀀋erent prime numbers p and q. Which of the following must be true?  \nI. N is not a prime number.  \nII. N is divisible by pq.  \nIII. N is an odd integer.  \n(A) I only  \n(B) II only  \n(C) I and II only  \n(D) I and III only  \n(E) I, II, and III  \n7. A perfect square is an integer that is the square of an integer. Suppose that m and n are positive integers such that mn > 15. If 15mn is a perfect square, what is the least possible value of mn?  \n8. M is a set of six consecutive even integers. When the least three integers of set M are summed, the result is x. When the greatest three integers of set M are summed, the result is y. Which of the following is true?  \n(A) y = x − 18  \n(B) y = x + 18  \n(C) y = 2x  \n(D) y = 2x + 4  \n(E) y = 2x + 6  \n[erikthered.com/tutor](erikthered.com/tutor) pg. 1  \nSAT Math Hard Practice Quiz  \n9. A three-digit number, XYZ, is formed of three di􀀋erent non-zero digits X , Y , and Z. A new number is formed by rearranging the same three digits. What is the greatest possible di􀀋erence between the two numbers? (For example, 345 could be rearranged into 435, for a di􀀋erence of 435 − 345 = 90.)  \n10. An integer is subtracted from its square. The result could be which of the following?  \n(A) A negative integer.  \n(B) An odd integer.  \n(C) The product of two consecutive even integers.  \n(D) The product of two consecutive odd integers.  \n(E) The product of two consecutive integers.  \n[erikthered.com/tutor](erikthered.com/tutor) pg. 2  \nSAT Math Hard Practice Quiz  \nAlgebra and Functions  \n1. Let m be an even integer. How many possible values of  \nm satisfy √m + 7 ≤ 3 ?  \n(A) One  \n(B) Two  \n(C) Three  \n(D) Four  \n(E) Five  \n2. Let x be de􀀌ned by x = xx~~ ~~~~ ~~31 for any x such that x  1. Which of the following is equivalent to x − 1 ?  \nx + 2 (A)  \nx − 1  \n 4   \n(B)  \nx − 1  \n2x + 4  \n(C)  \nx − 1  \n 2   \n(D)  \nx − 1  \nx + 2  \n(E)  \nx − 2  \n3. Let a and b be numbers such that a3 = b2 . Which of the  \nfollowing is equivalent to b √a ?  \n(A) b2/3  \n(B) b4/3  \n(C) b2  \n(D) b3  \n(E) b4  \n4. Let m and n be positive integers such that one-third of m is n less than one-half of m. Which of the following is a possible value of m?  \n(A) 15  \n(B) 21  \n(C) 24  \n(D) 26  \n(E) 28  \n5. If a and b are numbers such that (a−4)(b+6) = 0, then what is the smallest possible value of a2 + b2 ?  \n6. Let f (x) = ax2 and g (x) = bx4 for any value of x. If a and b are positive constants, for how many values of x is f (x) = g(x)?  \n(A) None  \n(B) One  \n(C) Two  \n(D) Three  \n(E) Four  \n7. Let a and b be numbers such that 30 \u003C a \u003C 40 and 50 \u003C b \u003C 70. Which of the following represents all possible values of a − b ?  \n(A) −40 \u003C a − b \u003C −20  \n(B) −40 \u003C a − b \u003C −10  \n(C) −30 \u003C a − b \u003C −20  \n(D) −20 \u003C a − b \u003C −10  \n(E) −20 \u003C a − b \u003C 30  \n[erikthered.com/tutor](erikthered.com/tutor) pg. 3  \nx3 + y12 = z  \n8. In the equation shown above, x, y, and z are positive integers. All of the following could be a possible valu","cbCaiviilfetgd7A","https://ap.wps.com/l/cbCaiviilfetgd7A","pdf",155798,18,"English","# Numbers and Operations\n## Algebra and Functions\n## Geometry","[{\"question\":\"What types of number theory concepts appear in the Numbers and Operations section?\",\"answer\":\"The section includes ratios with removals, monotonic digit integers, prime power factor counting, divisibility by two primes, perfect-square constraints, and relationships among consecutive even integers.\"},{\"question\":\"How do the sequence and rearrangement problems test SAT skills?\",\"answer\":\"The sequence question uses a defined starting term and a constant step to compute a specific term, while the rearrangement question maximizes the difference between two numbers formed from the same nonzero digits.\"},{\"question\":\"What kinds of algebra and functions topics are covered after Numbers and Operations?\",\"answer\":\"The Algebra and Functions section includes inequality solving for integer parameters, equivalent-expression transformations, simplifying radicals involving exponents, linear range reasoning, solving for zeros via product equations, and function equality/graph symmetry questions.\"}]","SAT Math Hard Practice Quiz - Numbers and Operations | PDF",45]