{"id":155,"date":"2020-01-19T16:01:20","date_gmt":"2020-01-19T16:01:20","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=155"},"modified":"2020-10-12T16:06:42","modified_gmt":"2020-10-12T16:06:42","slug":"mathcounts-exercises-14","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=155","title":{"rendered":"MATHCOUNTS Exercises &#8211; 14"},"content":{"rendered":"\n<p>Lisa wants to use her calculator to square a two-digit positive integer, but she accidentally enters the tens digit incorrectly. When she squares the number entered, the result is 2340 greater than the result she would have gotten had she correctly entered the tens digit. What is the sum of the two-digit number Lisa entered and the two-digit number she meant to enter? <em>Source: 2019 MATHCOUNTS Chapter Sprint Round. <\/em> <a onclick='toggle_visibility(\"mathcounts-exercises-14-solution\");'>Click here for the solution.<\/a><\/p>\n<div id=\"mathcounts-exercises-14-solution\" style=\"display:none\">\n<p><strong>Solution<\/strong>\nLets assume the bigger 2-digit number is \\(10x+y\\), and the smaller 2-digit number is \\(10z+y\\), we have:\n$$(10x + y)^2-(10z+y)^2=2340$$\n$$100x^2+20xy+y^2-(100z^2+20zy+y^2)=2340$$\nSimplify the above equation, we have:\n$$(x-z)(5x+y+5z)=117=1\\cdot 3\\cdot 3\\cdot 13$$ $$ \\because 10 > x > z > 0 $$ $$\\therefore x-z=1 \\ \\ or \\ \\ x-z=3 $$\nIf \\(x-z=1\\), we have\n$$ 5x + y + 5z=117 $$\nwhich is impossible, as the maximum possible value of \\(5x+y+5z\\) is 99. Therefore,\n$$x-z=3\\tag{1}$$ $$5x+y+5z=39\\tag{2}$$\nFrom (1), we have\n$$x=z+3\\tag{3}$$\nBased on (2) and (3), we have\n$$5(z+3)+y+5z=39$$\nSimplify the above equation, we have\n$$y+10z=24$$\nSince \\(y\\) and \\(z\\) are single digit numbers, the only solution is\n$$ y=4 \\ \\ \\ and \\ \\ \\ z=2$$\nTherefore \\(x=5\\). The sum of \\(10x+y\\) and \\(10z+y\\) is\n$$ 54+24=78$$\n<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Lisa wants to use her calculator to square a two-digit positive integer, but she accidentally enters the tens digit incorrectly. When she squares the number entered, the result is 2340 greater than the result she would have gotten had she &hellip; <a href=\"https:\/\/mathfun4kids.com\/mlog\/?p=155\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false},"categories":[13,11],"tags":[],"_links":{"self":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/155"}],"collection":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=155"}],"version-history":[{"count":1,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/155\/revisions"}],"predecessor-version":[{"id":156,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/155\/revisions\/156"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=155"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=155"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=155"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}