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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Clock angle problem</span></span>
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<p><b>Clock angle problems</b> are a type of <a href="Mathematical_problem" title="Mathematical problem">mathematical problem</a> which involve finding the angle between the hands of an <a href="Analog_clock" class="mw-redirect" title="Analog clock">analog clock</a>.
</p>
<meta property="mw:PageProp/toc">
<div class="mw-heading mw-heading2"><h2 id="Math_problem">Math problem</h2></div>
<p>Clock angle problems relate two different measurements: <a href="Angle" title="Angle">angles</a> and <a href="Time" title="Time">time</a>. The angle is typically measured in <a href="Degree_(angle)" title="Degree (angle)">degrees</a> from the mark of number 12 clockwise. The time is usually based on a <a href="12-hour_clock" title="12-hour clock">12-hour clock</a>.
</p><p>A method to solve such problems is to consider the rate of change of the angle in degrees per minute. The hour hand of a normal 12-hour analogue clock turns 360° in 12 hours (720 minutes) or 0.5° per minute. The minute hand rotates through 360° in 60 minutes or 6° per minute.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading3"><h3 id="Equation_for_the_angle_of_the_hour_hand">Equation for the angle of the hour hand</h3></div>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{\text{hr}}=0.5^{\circ }\times M_{\Sigma }=0.5^{\circ }\times (60\times H+M)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>θ<!-- θ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>hr</mtext>
</mrow>
</msub>
<mo>=</mo>
<msup>
<mn>0.5</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>∘<!-- ∘ --></mo>
</mrow>
</msup>
<mo>×<!-- × --></mo>
<msub>
<mi>M</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">Σ<!-- Σ --></mi>
</mrow>
</msub>
<mo>=</mo>
<msup>
<mn>0.5</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>∘<!-- ∘ --></mo>
</mrow>
</msup>
<mo>×<!-- × --></mo>
<mo stretchy="false">(</mo>
<mn>60</mn>
<mo>×<!-- × --></mo>
<mi>H</mi>
<mo>+</mo>
<mi>M</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \theta _{\text{hr}}=0.5^{\circ }\times M_{\Sigma }=0.5^{\circ }\times (60\times H+M)}</annotation>
</semantics>
</math></span><img src="./edf7c126ebf7ec6825123ba6b5682ff6732ed020.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.805ex; height:2.843ex;" alt="{\displaystyle \theta _{\text{hr}}=0.5^{\circ }\times M_{\Sigma }=0.5^{\circ }\times (60\times H+M)}" loading="lazy"></span></dd></dl>
<p>where:
</p>
<ul><li><span class="texhtml mvar" style="font-style:italic;">θ</span> is the angle in degrees of the hand measured clockwise from the 12</li>
<li><span class="texhtml mvar" style="font-style:italic;">H</span> is the hour.</li>
<li><span class="texhtml mvar" style="font-style:italic;">M</span> is the minutes past the hour.</li>
<li><span class="texhtml"><i>M</i><sub>Σ</sub></span> is the number of minutes since 12 o'clock. <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{\Sigma }=(60\times H+M)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>M</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">Σ<!-- Σ --></mi>
</mrow>
</msub>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mn>60</mn>
<mo>×<!-- × --></mo>
<mi>H</mi>
<mo>+</mo>
<mi>M</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle M_{\Sigma }=(60\times H+M)}</annotation>
</semantics>
</math></span><img src="./3e8da0d0c8c6bbe905beec87f243325054994ba9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.092ex; height:2.843ex;" alt="{\displaystyle M_{\Sigma }=(60\times H+M)}" loading="lazy"></span></li></ul>
<div class="mw-heading mw-heading3"><h3 id="Equation_for_the_angle_of_the_minute_hand">Equation for the angle of the minute hand</h3></div>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{\text{min.}}=6^{\circ }\times M}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>θ<!-- θ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>min.</mtext>
</mrow>
</msub>
<mo>=</mo>
<msup>
<mn>6</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>∘<!-- ∘ --></mo>
</mrow>
</msup>
<mo>×<!-- × --></mo>
<mi>M</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \theta _{\text{min.}}=6^{\circ }\times M}</annotation>
</semantics>
</math></span><img src="./de2db3677d7830c9f0ce7224d190397979daeedb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.118ex; height:2.676ex;" alt="{\displaystyle \theta _{\text{min.}}=6^{\circ }\times M}" loading="lazy"></span></dd></dl>
<p>where:
</p>
<ul><li><span class="texhtml mvar" style="font-style:italic;">θ</span> is the angle in degrees of the hand measured clockwise from the 12 o'clock position.</li>
<li><span class="texhtml mvar" style="font-style:italic;">M</span> is the minute.</li></ul>
<div class="mw-heading mw-heading4"><h4 id="Example">Example</h4></div>
<p>The time is 5:24. The angle in degrees of the hour hand is:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{\text{hr}}=0.5^{\circ }\times (60\times 5+24)=162^{\circ }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>θ<!-- θ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>hr</mtext>
</mrow>
</msub>
<mo>=</mo>
<msup>
<mn>0.5</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>∘<!-- ∘ --></mo>
</mrow>
</msup>
<mo>×<!-- × --></mo>
<mo stretchy="false">(</mo>
<mn>60</mn>
<mo>×<!-- × --></mo>
<mn>5</mn>
<mo>+</mo>
<mn>24</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msup>
<mn>162</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>∘<!-- ∘ --></mo>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \theta _{\text{hr}}=0.5^{\circ }\times (60\times 5+24)=162^{\circ }}</annotation>
</semantics>
</math></span><img src="./1b7e87a82ccb1c512be602fac7c90d9549d18388.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.789ex; height:2.843ex;" alt="{\displaystyle \theta _{\text{hr}}=0.5^{\circ }\times (60\times 5+24)=162^{\circ }}" loading="lazy"></span></dd></dl>
<p>The angle in degrees of the minute hand is:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{\text{min.}}=6^{\circ }\times 24=144^{\circ }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>θ<!-- θ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>min.</mtext>
</mrow>
</msub>
<mo>=</mo>
<msup>
<mn>6</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>∘<!-- ∘ --></mo>
</mrow>
</msup>
<mo>×<!-- × --></mo>
<mn>24</mn>
<mo>=</mo>
<msup>
<mn>144</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>∘<!-- ∘ --></mo>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \theta _{\text{min.}}=6^{\circ }\times 24=144^{\circ }}</annotation>
</semantics>
</math></span><img src="./f10c3d92e38c76bb01c53b6694bad210de2d73c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.641ex; height:2.676ex;" alt="{\displaystyle \theta _{\text{min.}}=6^{\circ }\times 24=144^{\circ }}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading3"><h3 id="Equation_for_the_angle_between_the_hands">Equation for the angle between the hands</h3></div>
<p>The angle between the hands can be found using the following formula:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\Delta \theta &=\vert \theta _{\text{hr}}-\theta _{\text{min.}}\vert \\&=\vert 0.5^{\circ }\times (60\times H+M)-6^{\circ }\times M\vert \\&=\vert 0.5^{\circ }\times (60\times H+M)-0.5^{\circ }\times 12\times M\vert \\&=\vert 0.5^{\circ }\times (60\times H-11\times M)\vert \\\end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
<mtr>
<mtd>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>θ<!-- θ --></mi>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mo fence="false" stretchy="false">|</mo>
<msub>
<mi>θ<!-- θ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>hr</mtext>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>θ<!-- θ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>min.</mtext>
</mrow>
</msub>
<mo fence="false" stretchy="false">|</mo>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mo fence="false" stretchy="false">|</mo>
<msup>
<mn>0.5</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>∘<!-- ∘ --></mo>
</mrow>
</msup>
<mo>×<!-- × --></mo>
<mo stretchy="false">(</mo>
<mn>60</mn>
<mo>×<!-- × --></mo>
<mi>H</mi>
<mo>+</mo>
<mi>M</mi>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<msup>
<mn>6</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>∘<!-- ∘ --></mo>
</mrow>
</msup>
<mo>×<!-- × --></mo>
<mi>M</mi>
<mo fence="false" stretchy="false">|</mo>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mo fence="false" stretchy="false">|</mo>
<msup>
<mn>0.5</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>∘<!-- ∘ --></mo>
</mrow>
</msup>
<mo>×<!-- × --></mo>
<mo stretchy="false">(</mo>
<mn>60</mn>
<mo>×<!-- × --></mo>
<mi>H</mi>
<mo>+</mo>
<mi>M</mi>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<msup>
<mn>0.5</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>∘<!-- ∘ --></mo>
</mrow>
</msup>
<mo>×<!-- × --></mo>
<mn>12</mn>
<mo>×<!-- × --></mo>
<mi>M</mi>
<mo fence="false" stretchy="false">|</mo>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mo fence="false" stretchy="false">|</mo>
<msup>
<mn>0.5</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>∘<!-- ∘ --></mo>
</mrow>
</msup>
<mo>×<!-- × --></mo>
<mo stretchy="false">(</mo>
<mn>60</mn>
<mo>×<!-- × --></mo>
<mi>H</mi>
<mo>−<!-- − --></mo>
<mn>11</mn>
<mo>×<!-- × --></mo>
<mi>M</mi>
<mo stretchy="false">)</mo>
<mo fence="false" stretchy="false">|</mo>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\Delta \theta &=\vert \theta _{\text{hr}}-\theta _{\text{min.}}\vert \\&=\vert 0.5^{\circ }\times (60\times H+M)-6^{\circ }\times M\vert \\&=\vert 0.5^{\circ }\times (60\times H+M)-0.5^{\circ }\times 12\times M\vert \\&=\vert 0.5^{\circ }\times (60\times H-11\times M)\vert \\\end{aligned}}}</annotation>
</semantics>
</math></span><img src="./f8f33d394dbcbe0d4c92a86e190957585b1dddd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:46.671ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}\Delta \theta &=\vert \theta _{\text{hr}}-\theta _{\text{min.}}\vert \\&=\vert 0.5^{\circ }\times (60\times H+M)-6^{\circ }\times M\vert \\&=\vert 0.5^{\circ }\times (60\times H+M)-0.5^{\circ }\times 12\times M\vert \\&=\vert 0.5^{\circ }\times (60\times H-11\times M)\vert \\\end{aligned}}}" loading="lazy"></span></dd></dl>
<p>where
</p>
<ul><li><span class="texhtml mvar" style="font-style:italic;">H</span> is the hour</li>
<li><span class="texhtml mvar" style="font-style:italic;">M</span> is the minute</li></ul>
<p>If the angle is greater than 180 degrees then subtract it from 360 degrees.
</p>
<div class="mw-heading mw-heading4"><h4 id="Example_1">Example 1</h4></div>
<p>The time is 2:20.
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\Delta \theta &=\vert 0.5^{\circ }\times (60\times 2-11\times 20)\vert \\&=\vert 0.5^{\circ }\times (120-220)\vert \\&=50^{\circ }\end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
<mtr>
<mtd>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>θ<!-- θ --></mi>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mo fence="false" stretchy="false">|</mo>
<msup>
<mn>0.5</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>∘<!-- ∘ --></mo>
</mrow>
</msup>
<mo>×<!-- × --></mo>
<mo stretchy="false">(</mo>
<mn>60</mn>
<mo>×<!-- × --></mo>
<mn>2</mn>
<mo>−<!-- − --></mo>
<mn>11</mn>
<mo>×<!-- × --></mo>
<mn>20</mn>
<mo stretchy="false">)</mo>
<mo fence="false" stretchy="false">|</mo>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mo fence="false" stretchy="false">|</mo>
<msup>
<mn>0.5</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>∘<!-- ∘ --></mo>
</mrow>
</msup>
<mo>×<!-- × --></mo>
<mo stretchy="false">(</mo>
<mn>120</mn>
<mo>−<!-- − --></mo>
<mn>220</mn>
<mo stretchy="false">)</mo>
<mo fence="false" stretchy="false">|</mo>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<msup>
<mn>50</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>∘<!-- ∘ --></mo>
</mrow>
</msup>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\Delta \theta &=\vert 0.5^{\circ }\times (60\times 2-11\times 20)\vert \\&=\vert 0.5^{\circ }\times (120-220)\vert \\&=50^{\circ }\end{aligned}}}</annotation>
</semantics>
</math></span><img src="./cc54114d4b20e419151751f933408df2f19b44ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:33.504ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}\Delta \theta &=\vert 0.5^{\circ }\times (60\times 2-11\times 20)\vert \\&=\vert 0.5^{\circ }\times (120-220)\vert \\&=50^{\circ }\end{aligned}}}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading4"><h4 id="Example_2">Example 2</h4></div>
<p>The time is 10:16.
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\Delta \theta &=\vert 0.5^{\circ }\times (60\times 10-11\times 16)\vert \\&=\vert 0.5^{\circ }\times (600-176)\vert \\&=212^{\circ }\ \ (>180^{\circ })\\&=360^{\circ }-212^{\circ }\\&=148^{\circ }\end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
<mtr>
<mtd>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>θ<!-- θ --></mi>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mo fence="false" stretchy="false">|</mo>
<msup>
<mn>0.5</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>∘<!-- ∘ --></mo>
</mrow>
</msup>
<mo>×<!-- × --></mo>
<mo stretchy="false">(</mo>
<mn>60</mn>
<mo>×<!-- × --></mo>
<mn>10</mn>
<mo>−<!-- − --></mo>
<mn>11</mn>
<mo>×<!-- × --></mo>
<mn>16</mn>
<mo stretchy="false">)</mo>
<mo fence="false" stretchy="false">|</mo>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mo fence="false" stretchy="false">|</mo>
<msup>
<mn>0.5</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>∘<!-- ∘ --></mo>
</mrow>
</msup>
<mo>×<!-- × --></mo>
<mo stretchy="false">(</mo>
<mn>600</mn>
<mo>−<!-- − --></mo>
<mn>176</mn>
<mo stretchy="false">)</mo>
<mo fence="false" stretchy="false">|</mo>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<msup>
<mn>212</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>∘<!-- ∘ --></mo>
</mrow>
</msup>
<mtext> </mtext>
<mtext> </mtext>
<mo stretchy="false">(</mo>
<mo>></mo>
<msup>
<mn>180</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>∘<!-- ∘ --></mo>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<msup>
<mn>360</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>∘<!-- ∘ --></mo>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<msup>
<mn>212</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>∘<!-- ∘ --></mo>
</mrow>
</msup>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<msup>
<mn>148</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>∘<!-- ∘ --></mo>
</mrow>
</msup>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\Delta \theta &=\vert 0.5^{\circ }\times (60\times 10-11\times 16)\vert \\&=\vert 0.5^{\circ }\times (600-176)\vert \\&=212^{\circ }\ \ (>180^{\circ })\\&=360^{\circ }-212^{\circ }\\&=148^{\circ }\end{aligned}}}</annotation>
</semantics>
</math></span><img src="./41926dfdc107564ab5657f414d2e32dbfd92394e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.005ex; width:34.666ex; height:15.176ex;" alt="{\displaystyle {\begin{aligned}\Delta \theta &=\vert 0.5^{\circ }\times (60\times 10-11\times 16)\vert \\&=\vert 0.5^{\circ }\times (600-176)\vert \\&=212^{\circ }\ \ (>180^{\circ })\\&=360^{\circ }-212^{\circ }\\&=148^{\circ }\end{aligned}}}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading3"><h3 id="When_are_the_hour_and_minute_hands_of_a_clock_superimposed?">When are the hour and minute hands of a clock superimposed?</h3></div>
<p>The hour and minute hands are superimposed only when their angle is the same.
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\theta _{\text{min}}&=\theta _{\text{hr}}\\\Rightarrow 6^{\circ }\times M&=0.5^{\circ }\times (60\times H+M)\\\Rightarrow 12\times M&=60\times H+M\\\Rightarrow 11\times M&=60\times H\\\Rightarrow M&={\frac {60}{11}}\times H\\\Rightarrow M&=5.{\overline {45}}\times H\end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
<mtr>
<mtd>
<msub>
<mi>θ<!-- θ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>min</mtext>
</mrow>
</msub>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<msub>
<mi>θ<!-- θ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>hr</mtext>
</mrow>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<mo stretchy="false">⇒<!-- ⇒ --></mo>
<msup>
<mn>6</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>∘<!-- ∘ --></mo>
</mrow>
</msup>
<mo>×<!-- × --></mo>
<mi>M</mi>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<msup>
<mn>0.5</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>∘<!-- ∘ --></mo>
</mrow>
</msup>
<mo>×<!-- × --></mo>
<mo stretchy="false">(</mo>
<mn>60</mn>
<mo>×<!-- × --></mo>
<mi>H</mi>
<mo>+</mo>
<mi>M</mi>
<mo stretchy="false">)</mo>
</mtd>
</mtr>
<mtr>
<mtd>
<mo stretchy="false">⇒<!-- ⇒ --></mo>
<mn>12</mn>
<mo>×<!-- × --></mo>
<mi>M</mi>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mn>60</mn>
<mo>×<!-- × --></mo>
<mi>H</mi>
<mo>+</mo>
<mi>M</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mo stretchy="false">⇒<!-- ⇒ --></mo>
<mn>11</mn>
<mo>×<!-- × --></mo>
<mi>M</mi>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mn>60</mn>
<mo>×<!-- × --></mo>
<mi>H</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mo stretchy="false">⇒<!-- ⇒ --></mo>
<mi>M</mi>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>60</mn>
<mn>11</mn>
</mfrac>
</mrow>
<mo>×<!-- × --></mo>
<mi>H</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mo stretchy="false">⇒<!-- ⇒ --></mo>
<mi>M</mi>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mn>5.</mn>
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mn>45</mn>
<mo accent="false">¯<!-- ¯ --></mo>
</mover>
</mrow>
<mo>×<!-- × --></mo>
<mi>H</mi>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\theta _{\text{min}}&=\theta _{\text{hr}}\\\Rightarrow 6^{\circ }\times M&=0.5^{\circ }\times (60\times H+M)\\\Rightarrow 12\times M&=60\times H+M\\\Rightarrow 11\times M&=60\times H\\\Rightarrow M&={\frac {60}{11}}\times H\\\Rightarrow M&=5.{\overline {45}}\times H\end{aligned}}}</annotation>
</semantics>
</math></span><img src="./d2c77ccc06df10dbd6939520c9d644d68b1b2a99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.838ex; width:35.614ex; height:20.843ex;" alt="{\displaystyle {\begin{aligned}\theta _{\text{min}}&=\theta _{\text{hr}}\\\Rightarrow 6^{\circ }\times M&=0.5^{\circ }\times (60\times H+M)\\\Rightarrow 12\times M&=60\times H+M\\\Rightarrow 11\times M&=60\times H\\\Rightarrow M&={\frac {60}{11}}\times H\\\Rightarrow M&=5.{\overline {45}}\times H\end{aligned}}}" loading="lazy"></span></dd></dl>
<p><span class="texhtml mvar" style="font-style:italic;">H</span> is an integer in the range 0–11. This gives times of: 0:00, 1:05.<span style="text-decoration:overline;">45</span>, 2:10.<span style="text-decoration:overline;">90</span>, 3:16.<span style="text-decoration:overline;">36</span>, 4:21.<span style="text-decoration:overline;">81</span>, 5:27.<span style="text-decoration:overline;">27</span>. 6:32.<span style="text-decoration:overline;">72</span>, 7:38.<span style="text-decoration:overline;">18</span>, 8:43.<span style="text-decoration:overline;">63</span>, 9:49.<span style="text-decoration:overline;">09</span>,
10:54.<span style="text-decoration:overline;">54</span>, and 12:00.
(0.<span style="text-decoration:overline;">45</span> minutes are exactly 27.<span style="text-decoration:overline;">27</span> seconds.)
</p>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Clock_position" title="Clock position">Clock position</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">
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.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}
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</style><cite id="CITEREFElgin,_Dave2007" class="citation journal cs1">Elgin, Dave (2007). "Angles on the Clock Face". <i>Mathematics in School</i>. <b>36</b> (5). The Mathematical Association: <span class="nowrap">4–</span>5. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/30216063">30216063</a>.</cite></span>
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<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>
<ul><li><a rel="nofollow" class="external free" href="https://web.archive.org/web/20100615083701/http://delphiforfun.org/Programs/clock_angle.htm">https://web.archive.org/web/20100615083701/http://delphiforfun.org/Programs/clock_angle.htm</a></li>
<li><a rel="nofollow" class="external free" href="http://www.ldlewis.com/hospital_clock/">http://www.ldlewis.com/hospital_clock/</a> - extensive clock angle analysis</li>
<li><a rel="nofollow" class="external free" href="https://web.archive.org/web/20100608044951/http://www.jimloy.com/puzz/clock1.htm">https://web.archive.org/web/20100608044951/http://www.jimloy.com/puzz/clock1.htm</a></li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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