{"id":4188,"date":"2025-06-09T14:30:30","date_gmt":"2025-06-09T14:30:30","guid":{"rendered":"https:\/\/diznr.com\/?p=4188"},"modified":"2025-06-09T14:30:30","modified_gmt":"2025-06-09T14:30:30","slug":"mathematics-short-tricks-mathematics-average-tricks","status":"publish","type":"post","link":"https:\/\/www.reilsolar.com\/pdf\/mathematics-short-tricks-mathematics-average-tricks\/","title":{"rendered":"Mathematics Short Tricks\/ MATHEMATICS AVERAGE TRICKS."},"content":{"rendered":"

Mathematics Short Tricks\/ MATHEMATICS AVERAGE TRICKS.<\/p>\n

[fvplayer id=”649″]<\/p>\n

Mathematics Short Tricks for Averages<\/strong><\/h3>\n

What is an Average?<\/strong><\/h3>\n

The average<\/strong> (or mean) of a set of numbers is the sum<\/strong> of the values divided by the number of values<\/strong>.<\/p>\n

Formula:<\/strong><\/p>\n

Average=Sum\u00a0of\u00a0all\u00a0valuesNumber\u00a0of\u00a0values\\text{Average} = \\frac{\\text{Sum of all values}}{\\text{Number of values}}<\/span>Average<\/span><\/span>=<\/span><\/span>Number\u00a0of\u00a0values<\/span>Sum of all values<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

Shortcut Tricks for Solving Average Problems:<\/strong><\/h3>\n

1. Shortcut for Finding the New Average:<\/strong><\/h3>\n

If the average<\/strong> of nn<\/span>n<\/span><\/span><\/span><\/span> numbers is A<\/strong>, and a new value x<\/strong> is added, the new average becomes:<\/p>\n

New\u00a0Average=A\u00d7n+xn+1\\text{New Average} = \\frac{A \\times n + x}{n + 1}<\/span>New\u00a0Average<\/span><\/span>=<\/span><\/span>n<\/span>+<\/span>1A<\/span>\u00d7<\/span>n<\/span>+<\/span>x<\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

Example:<\/strong>
Average of 5 numbers is 20<\/strong>. A new number 30<\/strong> is added. What is the new average?<\/p>\n

New\u00a0Average=20\u00d75+305+1=100+306=1306=21.67\\text{New Average} = \\frac{20 \\times 5 + 30}{5 + 1} = \\frac{100 + 30}{6} = \\frac{130}{6} = 21.67<\/span>New\u00a0Average<\/span><\/span>=<\/span><\/span>5+<\/span>120\u00d7<\/span>5+<\/span>30<\/span>\u200b<\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>6100+<\/span>30<\/span>\u200b<\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>6130<\/span>\u200b<\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>21.67<\/span><\/span><\/span><\/span><\/span><\/p>\n

2. Shortcut for Removing an Element:<\/strong><\/h3>\n

If a value x<\/strong> is removed<\/strong> from the set:<\/p>\n

New\u00a0Average=A\u00d7n\u2212xn\u22121\\text{New Average} = \\frac{A \\times n – x}{n – 1}<\/span>New\u00a0Average<\/span><\/span>=<\/span><\/span>n<\/span>\u2212<\/span>1A<\/span>\u00d7<\/span>n<\/span>\u2212<\/span>x<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

3. Shortcut for Consecutive Numbers:<\/strong><\/h3>\n

For consecutive numbers like 1,2,3,4,51, 2, 3, 4, 5<\/span>1<\/span>,<\/span>2<\/span>,<\/span>3<\/span>,<\/span>4<\/span>,<\/span>5<\/span><\/span><\/span><\/span>, the average is the middle number<\/strong>.<\/p>\n

Example:<\/strong>
Average of 4,5,6,7,84, 5, 6, 7, 8<\/span>4<\/span>,<\/span>5<\/span>,<\/span>6<\/span>,<\/span>7<\/span>,<\/span>8<\/span><\/span><\/span><\/span> is 6<\/strong> (middle number).<\/p>\n

4. Shortcut for Evenly Spaced Numbers:<\/strong><\/h3>\n

For evenly spaced numbers, the average is the average of the first and last numbers<\/strong>.<\/p>\n

Example:<\/strong>
Average of 10,20,30,4010, 20, 30, 40<\/span>10<\/span>,<\/span>20<\/span>,<\/span>30<\/span>,<\/span>40<\/span><\/span><\/span><\/span>:<\/p>\n

Average=10+402=25\\text{Average} = \\frac{10 + 40}{2} = 25<\/span>Average<\/span><\/span>=<\/span><\/span>210+<\/span>40<\/span>\u200b<\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>25<\/span><\/span><\/span><\/span><\/span><\/p>\n

5. Shortcut for Weighted Averages:<\/strong><\/h3>\n

When different groups have different averages and sizes:<\/p>\n

Weighted\u00a0Average=(A1\u00d7n1)+(A2\u00d7n2)n1+n2\\text{Weighted Average} = \\frac{(A_1 \\times n_1) + (A_2 \\times n_2)}{n_1 + n_2}<\/span>Weighted\u00a0Average<\/span><\/span>=<\/span><\/span>n<\/span>1<\/span><\/span>\u200b<\/span><\/span>+<\/span>n<\/span>2<\/span><\/span>\u200b<\/span><\/span>(<\/span>A<\/span>1<\/span><\/span>\u200b<\/span><\/span>\u00d7<\/span>n<\/span>1<\/span><\/span>\u200b<\/span><\/span>)<\/span>+<\/span>(<\/span>A<\/span>2<\/span><\/span>\u200b<\/span><\/span>\u00d7<\/span>n<\/span>2<\/span><\/span>\u200b<\/span><\/span>)<\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

Example:<\/strong>
Class A has an average of 60 for 20 students. Class B has an average of 70 for 30 students.<\/p>\n

Weighted\u00a0Average=(60\u00d720)+(70\u00d730)20+30=1200+210050=66\\text{Weighted Average} = \\frac{(60 \\times 20) + (70 \\times 30)}{20 + 30} = \\frac{1200 + 2100}{50} = 66<\/span>Weighted\u00a0Average<\/span><\/span>=<\/span><\/span>20+<\/span>30(<\/span>60\u00d7<\/span>20)<\/span>+<\/span>(<\/span>70\u00d7<\/span>30)<\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>501200+<\/span>2100<\/span>\u200b<\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>66<\/span><\/span><\/span><\/span><\/span><\/p>\n

Quick Tips:<\/strong><\/h3>\n
    \n
  1. If all values increase or decrease by the same amount, the average also increases or decreases by that amount.<\/li>\n
  2. If the same value is added or removed from all elements, the average remains unchanged.<\/li>\n<\/ol>\n

    Practice Questions:<\/strong><\/h3>\n
      \n
    1. The average of 6 numbers is 15. If a new number 21 is added, what is the new average?<\/li>\n
    2. The average of 5 consecutive even numbers is 16. Find the numbers.<\/li>\n
    3. The average of 10 numbers is 25. If one number, 30, is excluded, what is the new average?<\/li>\n<\/ol>\n

      If you need help solving these or more tricks, feel free to ask!<\/p>\n

      Mathematics Short Tricks\/ MATHEMATICS AVERAGE TRICKS.<\/a><\/h3>\n

      SSC CGL Tier 1 Maths Short Tricks and Formulas<\/a><\/h3>\n

      Here are some powerful short tricks for solving “Average” problems in Mathematics<\/strong>, useful for competitive exams like SSC, Banking, Railways, NEET, JEE (for general aptitude), and school-level exams:<\/p>\n

      \n

      \ud83d\udcda What is Average?<\/strong><\/h2>\n

      Formula:<\/strong><\/p>\n

      Average=Sum\u00a0of\u00a0all\u00a0observationsNumber\u00a0of\u00a0observations\\text{Average} = \\frac{\\text{Sum of all observations}}{\\text{Number of observations}}<\/span>Average<\/span><\/span>=<\/span><\/span>Number\u00a0of\u00a0observations<\/span>Sum\u00a0of\u00a0all\u00a0observations<\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

      \n

      \ud83d\udd22 Basic Tricks and Shortcuts for Averages<\/strong><\/h2>\n

      \u2705 1. Finding a New Average When a Value is Added or Removed<\/strong><\/h3>\n

      Trick:<\/strong><\/p>\n

      New\u00a0Average=Old\u00a0Average+(NewValue\u2212OldAverage)n+1\\text{New Average} = \\text{Old Average} + \\frac{(New Value – Old Average)}{n+1}<\/span>New\u00a0Average<\/span><\/span>=<\/span><\/span>Old\u00a0Average<\/span><\/span>+<\/span><\/span>n<\/span>+<\/span>1(<\/span>N<\/span>e<\/span>w<\/span>Va<\/span>l<\/span>u<\/span>e<\/span>\u2212<\/span>Ol<\/span>d<\/span>A<\/span>v<\/span>er<\/span>a<\/span>g<\/span>e<\/span>)<\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

      (where n<\/strong> is the number of items before addition)<\/p>\n

      \ud83e\udde0 Example:<\/strong><\/h3>\n

      Average of 5 numbers is 10. A new number 25 is added. Find the new average.
      Solution:<\/strong><\/p>\n

      New\u00a0Average=10+(25\u221210)6=10+156=10+2.5=12.5\\text{New Average} = 10 + \\frac{(25 – 10)}{6} = 10 + \\frac{15}{6} = 10 + 2.5 = 12.5<\/span>New\u00a0Average<\/span><\/span>=<\/span><\/span>10<\/span>+<\/span><\/span>6(<\/span>25\u2212<\/span>10)<\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>10<\/span>+<\/span><\/span>615<\/span>\u200b<\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>10<\/span>+<\/span><\/span>2.5<\/span>=<\/span><\/span>12.5<\/span><\/span><\/span><\/span><\/span><\/p>\n

      \n

      \u2705 2. Shortcut for Average Speed<\/strong><\/h3>\n

      If a person travels a certain distance at speed x<\/strong> and returns at speed y<\/strong>, then:<\/p>\n

      Average\u00a0Speed=2xyx+y\\text{Average Speed} = \\frac{2xy}{x + y}<\/span>Average\u00a0Speed<\/span><\/span>=<\/span><\/span>x<\/span>+<\/span>y<\/span>2x<\/span>y<\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

      \ud83e\udde0 Example:<\/strong><\/h3>\n

      A car travels to a city at 60 km\/h and returns at 40 km\/h. Find average speed.<\/p>\n

      Average\u00a0Speed=2\u00d760\u00d74060+40=4800100=48\u00a0km\/h\\text{Average Speed} = \\frac{2 \\times 60 \\times 40}{60 + 40} = \\frac{4800}{100} = 48 \\text{ km\/h}<\/span>Average\u00a0Speed<\/span><\/span>=<\/span><\/span>60+<\/span>402\u00d7<\/span>60\u00d7<\/span>40<\/span>\u200b<\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>1004800<\/span>\u200b<\/span><\/span><\/span><\/span><\/span>=<\/span><\/span>48<\/span>\u00a0km\/h<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

      \n

      \u2705 3. Shortcut for Finding a Missing Value<\/strong><\/h3>\n

      Trick:<\/strong><\/p>\n

      Missing\u00a0value=(Average\u00d7Total\u00a0items)\u2212(Sum\u00a0of\u00a0known\u00a0values)\\text{Missing value} = (\\text{Average} \\times \\text{Total items}) – (\\text{Sum of known values})<\/span>Missing\u00a0value<\/span><\/span>=<\/span><\/span>(<\/span>Average<\/span><\/span>\u00d7<\/span><\/span>Total\u00a0items<\/span><\/span>)<\/span>\u2212<\/span><\/span>(<\/span>Sum\u00a0of\u00a0known\u00a0values<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/p>\n

      \n

      \u2705 4. When Average of Subgroups is Known<\/strong><\/h3>\n

      Trick:<\/strong>
      If group A (avg = x, size = m) and group B (avg = y, size = n), then:<\/p>\n

      Combined\u00a0Average=(m\u00d7x)+(n\u00d7y)m+n\\text{Combined Average} = \\frac{(m \\times x) + (n \\times y)}{m + n}<\/span>Combined\u00a0Average<\/span><\/span>=<\/span><\/span>m<\/span>+<\/span>n<\/span>(<\/span>m<\/span>\u00d7<\/span>x<\/span>)<\/span>+<\/span>(<\/span>n<\/span>\u00d7<\/span>y<\/span>)<\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

      \n

      \u2705 5. Average of First n Natural Numbers<\/strong><\/h3>\n

      Average=n+12\\text{Average} = \\frac{n+1}{2}<\/span>Average<\/span><\/span>=<\/span><\/span>2n<\/span>+<\/span>1<\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

      \u2705 6. Average of Squares of First n Natural Numbers<\/strong><\/h3>\n

      Average=(n+1)(2n+1)6\\text{Average} = \\frac{(n+1)(2n+1)}{6}<\/span>Average<\/span><\/span>=<\/span><\/span>6(<\/span>n<\/span>+<\/span>1)<\/span>(<\/span>2n<\/span>+<\/span>1)<\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

      \n

      \ud83c\udfaf Memory Tricks \/ Tips<\/strong><\/h2>\n
        \n
      • \n

        “When one goes up, the average may go up \u2014 depending on how far it is from the mean.”<\/p>\n<\/li>\n

      • \n

        Keep shortcuts ready for average speed and number replacement.<\/p>\n<\/li>\n

      • \n

        Always check whether the new value added is above or below the current average.<\/p>\n<\/li>\n<\/ul>\n

        \n

        \ud83c\udfa5 Recommended Videos (Hindi + English Medium)<\/strong><\/h2>\n
          \n
        1. \n

          Maths Average Tricks by Abhinay Maths (Hindi)<\/strong><\/p>\n<\/li>\n

        2. \n

          Shortcuts for Averages by Dear Sir (Hindi)<\/strong><\/p>\n<\/li>\n

        3. \n

          Maths Tricks for Competitive Exams – Average<\/strong><\/p>\n<\/li>\n<\/ol>\n

          \n

          If you\u2019d like, I can create a printable PDF of these tricks<\/strong> or give you practice questions with answers<\/strong>. Would you like that?<\/p>\n","protected":false},"excerpt":{"rendered":"

          Mathematics Short Tricks\/ MATHEMATICS AVERAGE TRICKS. [fvplayer id=”649″] Mathematics Short Tricks for Averages What is an Average? The average (or mean) of a set of numbers is the sum of the values divided by the number of values. Formula: Average=Sum\u00a0of\u00a0all\u00a0valuesNumber\u00a0of\u00a0values\\text{Average} = \\frac{\\text{Sum of all values}}{\\text{Number of values}}Average=Number\u00a0of\u00a0valuesSum of all values Shortcut Tricks for Solving Average […]<\/p>\n","protected":false},"author":64,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[83],"tags":[],"class_list":["post-4188","post","type-post","status-publish","format-standard","hentry","category-mathematics-short-tricks"],"_links":{"self":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts\/4188","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/users\/64"}],"replies":[{"embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/comments?post=4188"}],"version-history":[{"count":0,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/posts\/4188\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/media?parent=4188"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/categories?post=4188"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.reilsolar.com\/pdf\/wp-json\/wp\/v2\/tags?post=4188"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}