{"id":23484,"date":"2025-06-17T16:27:58","date_gmt":"2025-06-17T16:27:58","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=23484"},"modified":"2025-06-17T16:28:00","modified_gmt":"2025-06-17T16:28:00","slug":"two-playful-kittens-working-together-can-unwind-all-the-balls-of-yarn-in-a-basket-9-minutes-faster-than-if-the-first-kitten-is-working-alone-and-4-minutes-faster-than-if-the-second-kitten-is-working","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/two-playful-kittens-working-together-can-unwind-all-the-balls-of-yarn-in-a-basket-9-minutes-faster-than-if-the-first-kitten-is-working-alone-and-4-minutes-faster-than-if-the-second-kitten-is-working\/","title":{"rendered":"Two playful kittens working together can unwind all the balls of yarn in a basket 9 minutes faster than if the first kitten is working alone, and 4 minutes faster than if the second kitten is working alone"},"content":{"rendered":"\n<p>Two playful kittens working together can unwind all the balls of yarn in a basket 9 minutes faster than if the first kitten is working alone, and 4 minutes faster than if the second kitten is working alone. How long does it take each kitten to unwind all the balls of yarn alone?<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\"><strong>The correct answer and explanation is:<\/strong><\/mark><\/p>\n\n\n\n<p>Let&#8217;s denote:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>xx = time (in minutes) it takes the <strong>first kitten<\/strong> to unwind all the yarn alone.<\/li>\n\n\n\n<li>yy = time (in minutes) it takes the <strong>second kitten<\/strong> to unwind all the yarn alone.<\/li>\n\n\n\n<li>tt = time (in minutes) it takes <strong>both kittens working together<\/strong> to unwind all the yarn.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Given information:<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Both kittens working together take tt minutes.<\/li>\n\n\n\n<li>Together they are 9 minutes faster than the first kitten alone: t=x\u22129t = x &#8211; 9<\/li>\n\n\n\n<li>Together they are 4 minutes faster than the second kitten alone: t=y\u22124t = y &#8211; 4<\/li>\n<\/ol>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 1: Express xx and yy in terms of tt<\/h3>\n\n\n\n<p>From the two equations above: x=t+9x = t + 9 y=t+4y = t + 4<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 2: Work rate setup<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>First kitten\u2019s work rate = 1x=1t+9\\frac{1}{x} = \\frac{1}{t+9} (jobs per minute)<\/li>\n\n\n\n<li>Second kitten\u2019s work rate = 1y=1t+4\\frac{1}{y} = \\frac{1}{t+4} (jobs per minute)<\/li>\n\n\n\n<li>Combined work rate = 1t\\frac{1}{t}<\/li>\n<\/ul>\n\n\n\n<p>Since their rates add when working together: 1t+9+1t+4=1t\\frac{1}{t+9} + \\frac{1}{t+4} = \\frac{1}{t}<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 3: Solve the equation<\/h3>\n\n\n\n<p>Multiply through by t(t+9)(t+4)t(t+9)(t+4) to clear denominators: t(t+4)+t(t+9)=(t+9)(t+4)t(t+4) + t(t+9) = (t+9)(t+4)<\/p>\n\n\n\n<p>Expand each term: t2+4t+t2+9t=t2+4t+9t+36t^2 + 4t + t^2 + 9t = t^2 + 4t + 9t + 36<\/p>\n\n\n\n<p>Simplify: 2t2+13t=t2+13t+362t^2 + 13t = t^2 + 13t + 36<\/p>\n\n\n\n<p>Subtract t2+13tt^2 + 13t from both sides: 2t2+13t\u2212t2\u221213t=362t^2 + 13t &#8211; t^2 &#8211; 13t = 36 t2=36t^2 = 36<\/p>\n\n\n\n<p>So, t=6(discard&nbsp;negative&nbsp;root&nbsp;t=\u22126)t = 6 \\quad (\\text{discard negative root } t = -6)<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Step 4: Find xx and yy<\/h3>\n\n\n\n<p>x=t+9=6+9=15&nbsp;minutesx = t + 9 = 6 + 9 = 15 \\text{ minutes} y=t+4=6+4=10&nbsp;minutesy = t + 4 = 6 + 4 = 10 \\text{ minutes}<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Answer:<\/strong><\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>First kitten alone takes <strong>15 minutes<\/strong>.<\/li>\n\n\n\n<li>Second kitten alone takes <strong>10 minutes<\/strong>.<\/li>\n\n\n\n<li>Together they take <strong>6 minutes<\/strong>.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation:<\/h3>\n\n\n\n<p>This problem involves rates of work. Each kitten has a rate of unwinding yarn defined as one job divided by the time it takes (jobs per minute). When working together, their rates add up to complete the job faster. The key was translating the problem into equations that describe their rates and how their combined work time compares to their individual times.<\/p>\n\n\n\n<p>By setting the time both kittens take together as tt, and expressing the individual times in terms of tt, we created an equation to solve for tt. The problem nicely shows how teamwork (combined rates) speeds up the process, and once tt was found, it was straightforward to find the individual times for each kitten.<\/p>\n\n\n\n<p>This approach can be used in many similar problems involving combined work, such as filling tanks, painting walls, or any situation where two or more workers (or machines) work together and finish faster than alone.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Two playful kittens working together can unwind all the balls of yarn in a basket 9 minutes faster than if the first kitten is working alone, and 4 minutes faster than if the second kitten is working alone. How long does it take each kitten to unwind all the balls of yarn alone? The correct [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-23484","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/23484","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=23484"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/23484\/revisions"}],"predecessor-version":[{"id":23485,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/23484\/revisions\/23485"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=23484"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=23484"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=23484"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}