Q: What is the ratio of the new area to the original area of the square if each side of the square is increased by 50%?  

<span class="pln">
</span><span class="typ">The</span><span class="pln"> ratio between the </span><span class="kwd">new</span><span class="pln"> area </span><span class="kwd">and</span><span class="pln"> the original area </span><span class="kwd">of</span><span class="pln"> the square can be found </span><span class="kwd">by</span><span class="pln"> </span><span class="kwd">using</span><span class="pln"> the formula </span><span class="kwd">for</span><span class="pln"> the area </span><span class="kwd">of</span><span class="pln"> a square </span><span class="kwd">and</span><span class="pln"> the percentage increase </span><span class="kwd">in</span><span class="pln"> the side length</span><span class="pun">.</span><span class="pln">

</span><span class="typ">The</span><span class="pln"> original area </span><span class="kwd">of</span><span class="pln"> the square </span><span class="kwd">is</span><span class="pln"> s</span><span class="pun">^</span><span class="lit">2</span><span class="pun">,</span><span class="pln"> </span><span class="kwd">where</span><span class="pln"> s </span><span class="kwd">is</span><span class="pln"> the side length</span><span class="pun">.</span><span class="pln"> </span><span class="typ">After</span><span class="pln"> the side length </span><span class="kwd">is</span><span class="pln"> increased </span><span class="kwd">by</span><span class="pln"> </span><span class="lit">50</span><span class="pun">%,</span><span class="pln"> the </span><span class="kwd">new</span><span class="pln"> side length </span><span class="kwd">is</span><span class="pln"> </span><span class="lit">1.5s</span><span class="pun">.</span><span class="pln">

</span><span class="typ">The</span><span class="pln"> </span><span class="kwd">new</span><span class="pln"> area </span><span class="kwd">of</span><span class="pln"> the square </span><span class="kwd">is</span><span class="pln"> </span><span class="pun">(</span><span class="lit">1.5s</span><span class="pun">)^</span><span class="lit">2</span><span class="pln"> </span><span class="pun">=</span><span class="pln"> </span><span class="lit">2.25s</span><span class="pun">^</span><span class="lit">2.</span><span class="pln">

</span><span class="typ">To</span><span class="pln"> find the ratio between the </span><span class="kwd">new</span><span class="pln"> area </span><span class="kwd">and</span><span class="pln"> the original area</span><span class="pun">,</span><span class="pln"> we divide the </span><span class="kwd">new</span><span class="pln"> area </span><span class="kwd">by</span><span class="pln"> the original area</span><span class="pun">:</span><span class="pln"> </span><span class="lit">2.25s</span><span class="pun">^</span><span class="lit">2</span><span class="pln"> </span><span class="pun">/</span><span class="pln"> s</span><span class="pun">^</span><span class="lit">2</span><span class="pln"> </span><span class="pun">=</span><span class="pln"> </span><span class="lit">2.25</span><span class="pun">.</span><span class="pln">

</span><span class="typ">Therefore</span><span class="pun">,</span><span class="pln"> the ratio between the </span><span class="kwd">new</span><span class="pln"> area </span><span class="kwd">and</span><span class="pln"> the original area </span><span class="kwd">of</span><span class="pln"> the square can be represented </span><span class="kwd">as</span><span class="pln"> </span><span class="lit">2.25</span><span class="pln"> </span><span class="pun">:</span><span class="pln"> </span><span class="lit">1</span><span class="pln"> which means </span><span class="kwd">for</span><span class="pln"> every </span><span class="lit">1</span><span class="pln"> unit </span><span class="kwd">of</span><span class="pln"> original area</span><span class="pun">,</span><span class="pln"> the </span><span class="kwd">new</span><span class="pln"> area </span><span class="kwd">is</span><span class="pln"> </span><span class="lit">2.25</span><span class="pln"> unit</span><span class="pun">.</span>