{"id":90981,"date":"2025-06-01T10:42:27","date_gmt":"2025-06-01T10:42:27","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=90981"},"modified":"2025-06-01T10:42:27","modified_gmt":"2025-06-01T10:42:27","slug":"the-sum-of-the-ages-of-a-and-b-in-years-is-22-the-product-of-their","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/the-sum-of-the-ages-of-a-and-b-in-years-is-22-the-product-of-their\/","title":{"rendered":"The sum of the ages of A and B (in years) is 22. The product of their"},"content":{"rendered":"

The sum of the ages of A and B (in years) is 22. The product of their ages two years back was 77. Which one of the following is the value of the difference of their current ages ?<\/p>\n

[amp_mcq option1=”2″ option2=”3″ option3=”4″ option4=”5″ correct=”option3″]<\/p>\n

\n
\n
This question was previously asked in<\/div>\n
UPSC CAPF – 2024<\/div>\n<\/div>\n
Download PDF<\/a>Attempt Online<\/a><\/div>\n<\/div>\n
\nThe correct option is C.
\n<\/section>\n
\nLet the current ages of A and B be ‘a’ and ‘b’ years, respectively.
\nGiven: a + b = 22.
\nTwo years back, their ages were (a-2) and (b-2).
\nGiven: (a-2)(b-2) = 77.
\nExpanding the second equation: ab – 2a – 2b + 4 = 77
\nab – 2(a + b) + 4 = 77.
\nSubstitute the first equation (a + b = 22) into this:
\nab – 2(22) + 4 = 77
\nab – 44 + 4 = 77
\nab – 40 = 77
\nab = 117.
\nWe need to find the difference of their current ages, which is |a – b|.
\nWe use the identity: (a – b)\u00b2 = (a + b)\u00b2 – 4ab.
\nSubstituting the values we have:
\n(a – b)\u00b2 = (22)\u00b2 – 4(117)
\n(a – b)\u00b2 = 484 – 468
\n(a – b)\u00b2 = 16.
\nTherefore, |a – b| = \u221a16 = 4.
\nThe difference in their current ages is 4 years.
\n<\/section>\n
\nWe can also find the individual ages. We have a + b = 22 and ab = 117. This means ‘a’ and ‘b’ are the roots of the quadratic equation x\u00b2 – (a+b)x + ab = 0, which is x\u00b2 – 22x + 117 = 0.
\nFactoring the quadratic: (x – 9)(x – 13) = 0.
\nThe roots are x = 9 and x = 13.
\nSo, the ages are 9 and 13 years.
\nLet’s check the conditions:
\nSum of ages = 9 + 13 = 22 (Correct).
\nAges two years back were 9-2 = 7 and 13-2 = 11.
\nProduct of ages two years back = 7 * 11 = 77 (Correct).
\nThe difference in their current ages is |13 – 9| = 4.
\n<\/section>\n","protected":false},"excerpt":{"rendered":"

The sum of the ages of A and B (in years) is 22. The product of their ages two years back was 77. Which one of the following is the value of the difference of their current ages ? [amp_mcq option1=”2″ option2=”3″ option3=”4″ option4=”5″ correct=”option3″] This question was previously asked in UPSC CAPF – 2024 … <\/p>\n

Detailed SolutionThe sum of the ages of A and B (in years) is 22. The product of their<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1085],"tags":[1103,1102],"class_list":["post-90981","post","type-post","status-publish","format-standard","hentry","category-upsc-capf","tag-1103","tag-quantitative-aptitude-and-reasoning","no-featured-image-padding"],"yoast_head":"\nThe sum of the ages of A and B (in years) is 22. The product of their<\/title>\n<meta name=\"description\" content=\"The correct option is C. Let the current ages of A and B be 'a' and 'b' years, respectively. Given: a + b = 22. Two years back, their ages were (a-2) and (b-2). Given: (a-2)(b-2) = 77. Expanding the second equation: ab - 2a - 2b + 4 = 77 ab - 2(a + b) + 4 = 77. Substitute the first equation (a + b = 22) into this: ab - 2(22) + 4 = 77 ab - 44 + 4 = 77 ab - 40 = 77 ab = 117. We need to find the difference of their current ages, which is |a - b|. We use the identity: (a - b)\u00b2 = (a + b)\u00b2 - 4ab. Substituting the values we have: (a - b)\u00b2 = (22)\u00b2 - 4(117) (a - b)\u00b2 = 484 - 468 (a - b)\u00b2 = 16. Therefore, |a - b| = \u221a16 = 4. The difference in their current ages is 4 years.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/exam.pscnotes.com\/mcq\/the-sum-of-the-ages-of-a-and-b-in-years-is-22-the-product-of-their\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"The sum of the ages of A and B (in years) is 22. The product of their\" \/>\n<meta property=\"og:description\" content=\"The correct option is C. Let the current ages of A and B be 'a' and 'b' years, respectively. Given: a + b = 22. Two years back, their ages were (a-2) and (b-2). Given: (a-2)(b-2) = 77. Expanding the second equation: ab - 2a - 2b + 4 = 77 ab - 2(a + b) + 4 = 77. Substitute the first equation (a + b = 22) into this: ab - 2(22) + 4 = 77 ab - 44 + 4 = 77 ab - 40 = 77 ab = 117. We need to find the difference of their current ages, which is |a - b|. We use the identity: (a - b)\u00b2 = (a + b)\u00b2 - 4ab. Substituting the values we have: (a - b)\u00b2 = (22)\u00b2 - 4(117) (a - b)\u00b2 = 484 - 468 (a - b)\u00b2 = 16. Therefore, |a - b| = \u221a16 = 4. The difference in their current ages is 4 years.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/the-sum-of-the-ages-of-a-and-b-in-years-is-22-the-product-of-their\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2025-06-01T10:42:27+00:00\" \/>\n<meta name=\"author\" content=\"rawan239\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"rawan239\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minute\" \/>\n<!-- \/ Yoast SEO Premium plugin. -->","yoast_head_json":{"title":"The sum of the ages of A and B (in years) is 22. The product of their","description":"The correct option is C. Let the current ages of A and B be 'a' and 'b' years, respectively. Given: a + b = 22. Two years back, their ages were (a-2) and (b-2). Given: (a-2)(b-2) = 77. Expanding the second equation: ab - 2a - 2b + 4 = 77 ab - 2(a + b) + 4 = 77. Substitute the first equation (a + b = 22) into this: ab - 2(22) + 4 = 77 ab - 44 + 4 = 77 ab - 40 = 77 ab = 117. We need to find the difference of their current ages, which is |a - b|. We use the identity: (a - b)\u00b2 = (a + b)\u00b2 - 4ab. Substituting the values we have: (a - b)\u00b2 = (22)\u00b2 - 4(117) (a - b)\u00b2 = 484 - 468 (a - b)\u00b2 = 16. Therefore, |a - b| = \u221a16 = 4. The difference in their current ages is 4 years.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/exam.pscnotes.com\/mcq\/the-sum-of-the-ages-of-a-and-b-in-years-is-22-the-product-of-their\/","og_locale":"en_US","og_type":"article","og_title":"The sum of the ages of A and B (in years) is 22. The product of their","og_description":"The correct option is C. Let the current ages of A and B be 'a' and 'b' years, respectively. Given: a + b = 22. Two years back, their ages were (a-2) and (b-2). Given: (a-2)(b-2) = 77. Expanding the second equation: ab - 2a - 2b + 4 = 77 ab - 2(a + b) + 4 = 77. Substitute the first equation (a + b = 22) into this: ab - 2(22) + 4 = 77 ab - 44 + 4 = 77 ab - 40 = 77 ab = 117. We need to find the difference of their current ages, which is |a - b|. We use the identity: (a - b)\u00b2 = (a + b)\u00b2 - 4ab. Substituting the values we have: (a - b)\u00b2 = (22)\u00b2 - 4(117) (a - b)\u00b2 = 484 - 468 (a - b)\u00b2 = 16. Therefore, |a - b| = \u221a16 = 4. The difference in their current ages is 4 years.","og_url":"https:\/\/exam.pscnotes.com\/mcq\/the-sum-of-the-ages-of-a-and-b-in-years-is-22-the-product-of-their\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2025-06-01T10:42:27+00:00","author":"rawan239","twitter_card":"summary_large_image","twitter_misc":{"Written by":"rawan239","Est. reading time":"1 minute"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/exam.pscnotes.com\/mcq\/the-sum-of-the-ages-of-a-and-b-in-years-is-22-the-product-of-their\/","url":"https:\/\/exam.pscnotes.com\/mcq\/the-sum-of-the-ages-of-a-and-b-in-years-is-22-the-product-of-their\/","name":"The sum of the ages of A and B (in years) is 22. The product of their","isPartOf":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#website"},"datePublished":"2025-06-01T10:42:27+00:00","dateModified":"2025-06-01T10:42:27+00:00","author":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209"},"description":"The correct option is C. Let the current ages of A and B be 'a' and 'b' years, respectively. Given: a + b = 22. Two years back, their ages were (a-2) and (b-2). Given: (a-2)(b-2) = 77. Expanding the second equation: ab - 2a - 2b + 4 = 77 ab - 2(a + b) + 4 = 77. Substitute the first equation (a + b = 22) into this: ab - 2(22) + 4 = 77 ab - 44 + 4 = 77 ab - 40 = 77 ab = 117. We need to find the difference of their current ages, which is |a - b|. We use the identity: (a - b)\u00b2 = (a + b)\u00b2 - 4ab. Substituting the values we have: (a - b)\u00b2 = (22)\u00b2 - 4(117) (a - b)\u00b2 = 484 - 468 (a - b)\u00b2 = 16. Therefore, |a - b| = \u221a16 = 4. The difference in their current ages is 4 years.","breadcrumb":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/the-sum-of-the-ages-of-a-and-b-in-years-is-22-the-product-of-their\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/exam.pscnotes.com\/mcq\/the-sum-of-the-ages-of-a-and-b-in-years-is-22-the-product-of-their\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/exam.pscnotes.com\/mcq\/the-sum-of-the-ages-of-a-and-b-in-years-is-22-the-product-of-their\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/exam.pscnotes.com\/mcq\/"},{"@type":"ListItem","position":2,"name":"UPSC CAPF","item":"https:\/\/exam.pscnotes.com\/mcq\/category\/upsc-capf\/"},{"@type":"ListItem","position":3,"name":"The sum of the ages of A and B (in years) is 22. 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