Often while working, an arithmetical answer is needed — exact is sometimes required, approximate is often acceptable — maybe the basis of a decision or a path forward is based on a few quick calculations, with more rigorous calculations to follow
Let’s illustrate with examples from two groups
First, we’ll learn to multiply by 11 by just writing down the answer
Question: What is
?
![Rendered by QuickLaTeX.com = \bold{72{,}459{,}596}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-fc92f702801010e649295064c4d72bef_l3.png)
Try it by yourself this time
![Rendered by QuickLaTeX.com \bold{6{,}587{,}236\cdot11}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-63a34b494d269f7e7b14fd06ec602c97_l3.png)
Solution:
![Rendered by QuickLaTeX.com = \bold{72{,}459{,}596}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-fc92f702801010e649295064c4d72bef_l3.png)
Try it by yourself this time
Remember the binomial theorem or Pascal’s Triangle. Let’s go back a few years and refresh.
Second, we’ll try a few exponent calculations
Question: What is
?
![Rendered by QuickLaTeX.com = 1+2(0.10)+(0.10)^2](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-d9d1d46193743486c29b31e431a91fee_l3.png)
![Rendered by QuickLaTeX.com \bold{1.10^2}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-a2a5cc1f78550895fd60c22cc5f92d13_l3.png)
Solution:
![Rendered by QuickLaTeX.com = 1+2(0.10)+(0.10)^2](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-d9d1d46193743486c29b31e431a91fee_l3.png)
![Rendered by QuickLaTeX.com = 1+0.20+0.01 = \bold{1.21}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-62c2c07e52d2e30c6a3daef95771b5a4_l3.png)
And
?
![Rendered by QuickLaTeX.com = 1+3(0.10)+3(0.10)^2+(0.10)^3](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-626db13ec5e4b6a390e75b2db3904f1d_l3.png)
![Rendered by QuickLaTeX.com = 1+0.300+0.030+0.001](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-0e94f36cb54f9a3c0e441faeb402c7e8_l3.png)
![Rendered by QuickLaTeX.com \bold{1.10^3}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-f78ee3ac705d2809732bdb1783ed7452_l3.png)
Solution:
![Rendered by QuickLaTeX.com = 1+3(0.10)+3(0.10)^2+(0.10)^3](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-626db13ec5e4b6a390e75b2db3904f1d_l3.png)
![Rendered by QuickLaTeX.com = 1+0.300+0.030+0.001](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-0e94f36cb54f9a3c0e441faeb402c7e8_l3.png)
![Rendered by QuickLaTeX.com = \bold{1.331}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-d1e1c564bdf5bbdb1d320f7990c0d747_l3.png)
Question: What is
?
![Rendered by QuickLaTeX.com 1+4(0.05)+6(0.05)^2+4(0.05)^3+(0.05)^4](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-32f4e4c93448f1f9ebaeaec6da6c7377_l3.png)
Now dropping the last two terms, we have
![Rendered by QuickLaTeX.com \approx 1+0.200000+6(0.002500) ...](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-6ccde22978bd16a94895889d9587514a_l3.png)
![Rendered by QuickLaTeX.com \approx 1+0.200000+ (0.015000) ...](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-cb3d737aff3d807652a1798f10ca4461_l3.png)
![Rendered by QuickLaTeX.com \bold{1.05^4}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-c3c067b15661fad879b4895d9c2eb175_l3.png)
Solution:
![Rendered by QuickLaTeX.com 1+4(0.05)+6(0.05)^2+4(0.05)^3+(0.05)^4](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-32f4e4c93448f1f9ebaeaec6da6c7377_l3.png)
Now dropping the last two terms, we have
![Rendered by QuickLaTeX.com \approx 1+0.200000+6(0.002500) ...](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-6ccde22978bd16a94895889d9587514a_l3.png)
![Rendered by QuickLaTeX.com \approx 1+0.200000+ (0.015000) ...](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-cb3d737aff3d807652a1798f10ca4461_l3.png)
![Rendered by QuickLaTeX.com \approx \bold{1.215}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-2fd60e72353a637aa2184f94bf6ecae2_l3.png)
The exact answer is ![Rendered by QuickLaTeX.com 1.21550625](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-ae6461963eff05e7e328d6c3dd0068eb_l3.png)
Therefore, by using only three terms we have an error of less than![Rendered by QuickLaTeX.com 0.05\%](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-4af62cdb0156ca097a100808a27bbc60_l3.png)
![Rendered by QuickLaTeX.com 1.21550625](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-ae6461963eff05e7e328d6c3dd0068eb_l3.png)
Therefore, by using only three terms we have an error of less than
![Rendered by QuickLaTeX.com 0.05\%](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-4af62cdb0156ca097a100808a27bbc60_l3.png)
Now let’s consider a few squares.
Question: What is
?
![Rendered by QuickLaTeX.com = 4900+6 \cdot 70+9](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-aad41fe897dcbf06eca7d56e3e6987c7_l3.png)
![Rendered by QuickLaTeX.com \bold{73^2}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-ebc536f7d9cd2ff6c61949b5e0ed364a_l3.png)
Recall that
Solution:
![Rendered by QuickLaTeX.com = 4900+6 \cdot 70+9](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-aad41fe897dcbf06eca7d56e3e6987c7_l3.png)
![Rendered by QuickLaTeX.com = 4900+420+9 = \bold{5329}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-0b7973fdeee5f4894369472775bf8860_l3.png)
Question: What is
?
![Rendered by QuickLaTeX.com = 8100-4 \cdot 90+4](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-8309c55587507c1039beef537e90ce47_l3.png)
![Rendered by QuickLaTeX.com = 8100-360+4](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-a850e87b6e22054600b3a9b42c2f7965_l3.png)
![Rendered by QuickLaTeX.com \bold{88^2}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-e8b39a69654949a909ef509dbad599cb_l3.png)
Recall that
Solution:
![Rendered by QuickLaTeX.com = 8100-4 \cdot 90+4](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-8309c55587507c1039beef537e90ce47_l3.png)
![Rendered by QuickLaTeX.com = 8100-360+4](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-a850e87b6e22054600b3a9b42c2f7965_l3.png)
![Rendered by QuickLaTeX.com = 8100-400+44 = \bold{7744}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-2593f763c1fb1d408b308cade3919942_l3.png)
Question: What is
?
![Rendered by QuickLaTeX.com = 1600-8 \cdot 40+4](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-2068466d4f3d0827f622d1c68511fd9f_l3.png)
![Rendered by QuickLaTeX.com = 1600-320+16](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-436ee445a374b6b144b373174cba6253_l3.png)
![Rendered by QuickLaTeX.com \bold{36^2}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-054753389b6fcf915c85b913dcd06854_l3.png)
Again
Solution:
![Rendered by QuickLaTeX.com = 1600-8 \cdot 40+4](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-2068466d4f3d0827f622d1c68511fd9f_l3.png)
![Rendered by QuickLaTeX.com = 1600-320+16](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-436ee445a374b6b144b373174cba6253_l3.png)
![Rendered by QuickLaTeX.com = 1600-400+96 = \bold{1296}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-7cfc7c80e1722cd48ed40820c3bc4885_l3.png)
Now let’s consider common temperature conversions.
Start with determining the Fahrenheit equivalent of a Centigrade temperature ?
Recall ![Rendered by QuickLaTeX.com \bold{F} = \frac{9}{5} \cdot \bold{C}+32](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-43873d0399c3a885179981ec7c9234a9_l3.png)
Or![Rendered by QuickLaTeX.com \bold{F} = [1.80] \cdot \bold{C}+32](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-a2717a95414a33c56933d4e8794042f2_l3.png)
Easier if approached using![Rendered by QuickLaTeX.com \bold{F} = [0.90 \cdot 2] \cdot \bold{C}+32](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-e674b40c13e38f9f25cd086e06f19427_l3.png)
Which is simply
less
plus ![Rendered by QuickLaTeX.com 32](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-22882aef9be2977d31de3bdcd09db94c_l3.png)
![Rendered by QuickLaTeX.com \bold{F} = \frac{9}{5} \cdot \bold{C}+32](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-43873d0399c3a885179981ec7c9234a9_l3.png)
Or
![Rendered by QuickLaTeX.com \bold{F} = [1.80] \cdot \bold{C}+32](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-a2717a95414a33c56933d4e8794042f2_l3.png)
Easier if approached using
![Rendered by QuickLaTeX.com \bold{F} = [0.90 \cdot 2] \cdot \bold{C}+32](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-e674b40c13e38f9f25cd086e06f19427_l3.png)
Which is simply
![Rendered by QuickLaTeX.com [2 \cdot \bold{C}]](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-e3cd1619c8a979524a3b349a171cd437_l3.png)
![Rendered by QuickLaTeX.com 10\%](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-b89fdbde1c2a875384f2142fb1afc121_l3.png)
![Rendered by QuickLaTeX.com 32](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-22882aef9be2977d31de3bdcd09db94c_l3.png)
Question: What is the Fahrenheit equivalent of
?
![Rendered by QuickLaTeX.com 350^\circ\text{C}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-7cc4ff702051c2746f5575fbcb12ab90_l3.png)
Say
less
plus ![Rendered by QuickLaTeX.com 32](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-22882aef9be2977d31de3bdcd09db94c_l3.png)
Say
plus ![Rendered by QuickLaTeX.com 32](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-22882aef9be2977d31de3bdcd09db94c_l3.png)
Say
plus ![Rendered by QuickLaTeX.com 32](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-22882aef9be2977d31de3bdcd09db94c_l3.png)
Therefore,![Rendered by QuickLaTeX.com \bold{662}^\circ\text{F}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-902552e50789a647917d61d244ddbeb7_l3.png)
![Rendered by QuickLaTeX.com [2 \cdot \bold{350}]](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-f1750b2b91d58de46b973f1c5abffa91_l3.png)
![Rendered by QuickLaTeX.com 10\%](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-b89fdbde1c2a875384f2142fb1afc121_l3.png)
![Rendered by QuickLaTeX.com 32](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-22882aef9be2977d31de3bdcd09db94c_l3.png)
Say
![Rendered by QuickLaTeX.com 700-70](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-5898438cfadfb47e5cdbbd53cc63c346_l3.png)
![Rendered by QuickLaTeX.com 32](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-22882aef9be2977d31de3bdcd09db94c_l3.png)
Say
![Rendered by QuickLaTeX.com 630](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-6b5bc7fca0cbf82878f989a7ebf57365_l3.png)
![Rendered by QuickLaTeX.com 32](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-22882aef9be2977d31de3bdcd09db94c_l3.png)
Therefore,
![Rendered by QuickLaTeX.com \bold{662}^\circ\text{F}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-902552e50789a647917d61d244ddbeb7_l3.png)
Recall ![Rendered by QuickLaTeX.com \bold{C} = \frac{5}{9} \cdot (\bold{F}-32)](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-94679dfa714d7f6a5b3b6178ada157e2_l3.png)
Or![Rendered by QuickLaTeX.com \bold{C} = [0.55\bar5] \cdot [\bold{F}-32]](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-d3b20eed2c1a2823cf1b52cfade697e2_l3.png)
Easier if approached using![Rendered by QuickLaTeX.com \bold{C} = [0.50+(0.1) \cdot 0.50+(0.01) \cdot 0.50] \cdot [\bold{F}-32]](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-e212b036b9f0ba94a1a719cde98801e9_l3.png)
So say
times half plus
times a tenth of a half and so forth
![Rendered by QuickLaTeX.com \bold{C} = \frac{5}{9} \cdot (\bold{F}-32)](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-94679dfa714d7f6a5b3b6178ada157e2_l3.png)
Or
![Rendered by QuickLaTeX.com \bold{C} = [0.55\bar5] \cdot [\bold{F}-32]](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-d3b20eed2c1a2823cf1b52cfade697e2_l3.png)
Easier if approached using
![Rendered by QuickLaTeX.com \bold{C} = [0.50+(0.1) \cdot 0.50+(0.01) \cdot 0.50] \cdot [\bold{F}-32]](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-e212b036b9f0ba94a1a719cde98801e9_l3.png)
So say
![Rendered by QuickLaTeX.com [\bold{F}-32]](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-f4a10f752e41a820279ad2ea606cec7c_l3.png)
![Rendered by QuickLaTeX.com [\bold{F}-32]](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-f4a10f752e41a820279ad2ea606cec7c_l3.png)
Question: What is the Centigrade equivalent of
?
![Rendered by QuickLaTeX.com 350^\circ\text{F}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-0994f8b10a49d33da3e0152fddc0c661_l3.png)
Say ![Rendered by QuickLaTeX.com [350-32] = 318](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-6f5477cf07d99ad8bd391ede7458ac7a_l3.png)
Say half of![Rendered by QuickLaTeX.com 318 = 159](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-10bce58d2c8cc657db4ef2a20af91465_l3.png)
Say![Rendered by QuickLaTeX.com 159+15.9 \text{ etc}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-5434e6119dcfa6f3e4265a4ff3e98e15_l3.png)
![Rendered by QuickLaTeX.com = 159+16-0.1+1.6+0.16](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-4b6dbbe49fb3e671a52c017918d21607_l3.png)
![Rendered by QuickLaTeX.com = 174.9+1.6+0.16 = 176.66](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-8e99c7a225e339692b7d0e01516337ea_l3.png)
Therefore,
.
![Rendered by QuickLaTeX.com [350-32] = 318](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-6f5477cf07d99ad8bd391ede7458ac7a_l3.png)
Say half of
![Rendered by QuickLaTeX.com 318 = 159](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-10bce58d2c8cc657db4ef2a20af91465_l3.png)
Say
![Rendered by QuickLaTeX.com 159+15.9 \text{ etc}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-5434e6119dcfa6f3e4265a4ff3e98e15_l3.png)
![Rendered by QuickLaTeX.com = 159+16-0.1+1.6+0.16](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-4b6dbbe49fb3e671a52c017918d21607_l3.png)
![Rendered by QuickLaTeX.com = 174.9+1.6+0.16 = 176.66](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-8e99c7a225e339692b7d0e01516337ea_l3.png)
Therefore,
![Rendered by QuickLaTeX.com \bold{176.66}^\circ\text{C}\approx \bold{177}^\circ\text{C}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-367e9897ecff065f510ea26e3715b84f_l3.png)
Question: What is the Centigrade equivalent of
?
![Rendered by QuickLaTeX.com 351^\circ\text{F}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-3ec44863d93b27d52400a5e003c4ed33_l3.png)
Say ![Rendered by QuickLaTeX.com [350-32] = 319 \text{ (odd number)}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-38625fbfeb50190d4ace2bb7a9adfbde_l3.png)
Say half of
with a remainder of ![Rendered by QuickLaTeX.com 1](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-4868771cbc422b5818f85500909ce433_l3.png)
Say![Rendered by QuickLaTeX.com 159+15.9 \text{ etc}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-5434e6119dcfa6f3e4265a4ff3e98e15_l3.png)
![Rendered by QuickLaTeX.com = 159+16-0.1+1.6+0.16+[0.6]](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-008d62a298547df5febdb5e699c4234a_l3.png)
![Rendered by QuickLaTeX.com = 176.66+[0.6]](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-1513b860cf525e04b46bd6820d82c529_l3.png)
![Rendered by QuickLaTeX.com [350-32] = 319 \text{ (odd number)}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-38625fbfeb50190d4ace2bb7a9adfbde_l3.png)
Say half of
![Rendered by QuickLaTeX.com 319 = 159](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-cecef48b424833c8522f45cb0ecab1c0_l3.png)
![Rendered by QuickLaTeX.com 1](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-4868771cbc422b5818f85500909ce433_l3.png)
Say
![Rendered by QuickLaTeX.com 159+15.9 \text{ etc}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-5434e6119dcfa6f3e4265a4ff3e98e15_l3.png)
![Rendered by QuickLaTeX.com = 159+16-0.1+1.6+0.16+[0.6]](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-008d62a298547df5febdb5e699c4234a_l3.png)
![Rendered by QuickLaTeX.com = 176.66+[0.6]](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-1513b860cf525e04b46bd6820d82c529_l3.png)
The extra
is to account for the remainder of 1
![Rendered by QuickLaTeX.com 0.6](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-6467b5d4c60ec605876c9aa484e54ddd_l3.png)
Therefore,
.
![Rendered by QuickLaTeX.com \bold{177.26}^\circ\text{C}\approx \bold{177}^\circ\text{C}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-188cd9f139be82f3bd050822a5c3be6b_l3.png)
Question: What is the Centigrade equivalent of
?
![Rendered by QuickLaTeX.com 1650^\circ\text{F}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-21e04ccfc7760b9864a01e285f83e733_l3.png)
Say ![Rendered by QuickLaTeX.com [1650-32] = 1618}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-c87c91ff219daed450c8ae8f088a57c3_l3.png)
Say half of![Rendered by QuickLaTeX.com 1618 = 809](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-1256820571f7533294940664237679f5_l3.png)
Say![Rendered by QuickLaTeX.com 159+15.9 \text{ etc}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-5434e6119dcfa6f3e4265a4ff3e98e15_l3.png)
![Rendered by QuickLaTeX.com = 809+80.9+8.1+0.8](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-64cdbb0f5a9671e34b9e9c7d806725a5_l3.png)
![Rendered by QuickLaTeX.com = 898.8](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-3248cc20a2d28beead72347367f43309_l3.png)
Therefore,
.
![Rendered by QuickLaTeX.com [1650-32] = 1618}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-c87c91ff219daed450c8ae8f088a57c3_l3.png)
Say half of
![Rendered by QuickLaTeX.com 1618 = 809](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-1256820571f7533294940664237679f5_l3.png)
Say
![Rendered by QuickLaTeX.com 159+15.9 \text{ etc}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-5434e6119dcfa6f3e4265a4ff3e98e15_l3.png)
![Rendered by QuickLaTeX.com = 809+80.9+8.1+0.8](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-64cdbb0f5a9671e34b9e9c7d806725a5_l3.png)
![Rendered by QuickLaTeX.com = 898.8](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-3248cc20a2d28beead72347367f43309_l3.png)
Therefore,
![Rendered by QuickLaTeX.com \bold{898.8}^\circ\text{C}\approx \bold{899}^\circ\text{C}\approx \bold{900}^\circ\text{C}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-6bf58ddce692c3aa812624c69afa260a_l3.png)
And last, Aliquot Parts
2, 5
![Rendered by QuickLaTeX.com 5 \cdot 2=10](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-48b02c4349ec3f71589fd776b93eab56_l3.png)
To multiply by 2, we can divide by 5 if it’s easier that way
2, 50
![Rendered by QuickLaTeX.com 50 \cdot 2=100](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-0c4de330ab30fcc81e69db97bf472ad5_l3.png)
4
![Rendered by QuickLaTeX.com 25 \cdot 4=100](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-a9729be59ccd983c6606b05b4f51aeac_l3.png)
![Rendered by QuickLaTeX.com 4 \cdot 25=100](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-b57699cd51ebf1d28f5b2a1b5f6b835f_l3.png)
To multiply by 25, we can divide by 4 if it’s easier that way
To divide by 25, we can multiply by 4 if it’s easier that way
8
![Rendered by QuickLaTeX.com 12.5 \cdot 8=100](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-4c8a5a0c9e607c8e545efddc19b024b6_l3.png)
![Rendered by QuickLaTeX.com 8 \cdot 12.5=100](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-6d9c2b2276ed5ba13f5c12c099c1446e_l3.png)
3
![Rendered by QuickLaTeX.com 33.33 \cdot 3=100](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-4baf642c986e25f8d5de6ee4ec9b1f12_l3.png)
![Rendered by QuickLaTeX.com 3 \cdot 33.33=100](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-74f25c39c426168443896e8e0592853b_l3.png)
6
![Rendered by QuickLaTeX.com 16.67 \cdot 6=16\frac{2}{3} \cdot 6=100](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-9255a7fa7b29000ce42b2a8152ed4c2c_l3.png)
![Rendered by QuickLaTeX.com 6 \cdot 16.67=6 \cdot 16\frac{2}{3}=100](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-091923c3d1cdbaa869e0291af680ebec_l3.png)
7
![Rendered by QuickLaTeX.com 14.29 \cdot 7=14\frac{2}{7} \cdot 7=100](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-e3d1a58a942a890de021f7ad50ba8b9a_l3.png)
![Rendered by QuickLaTeX.com 7 \cdot 14.29=12 \cdot 7\frac{2}{7}=100](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-ea116facba32d3dec26975843e184a06_l3.png)
9
![Rendered by QuickLaTeX.com 11.11 \cdot 9=11\frac{1}{9} \cdot 9=100](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-1ea3b288750a22be51aa461e0e377988_l3.png)
![Rendered by QuickLaTeX.com 9 \cdot 11.11=9 \cdot 11\frac{1}{9}=100](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-33e93213482c19d247989720790213d8_l3.png)
12
![Rendered by QuickLaTeX.com 8.33 \cdot 12=8\frac{1}{3} \cdot 12=100](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-afbde0cb94ab19005e3ecc99f55a5b4b_l3.png)
![Rendered by QuickLaTeX.com 12 \cdot 8.33=12 \cdot 8\frac{1}{3}=100](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-777e1695ff3b65d5b5400e14a3416234_l3.png)
15
![Rendered by QuickLaTeX.com 6.67 \cdot 15=6\frac{2}{3} \cdot 15=100](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-4a1a27b57d30641be4ebee1e7229bb2d_l3.png)
Click for a couple three quick examples using aliquot parts
How to use the divisors of 1, of 10, and of 100
2, 5
![Rendered by QuickLaTeX.com 5 \cdot 2=10](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-48b02c4349ec3f71589fd776b93eab56_l3.png)
![Rendered by QuickLaTeX.com 2 \cdot 5=10](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-d44e31f3f3035eaa089b12cb818c36df_l3.png)
To multiply by 5, we can divide by 2 and vice versa, whichever is easier!
What is
?
![Rendered by QuickLaTeX.com 5 \cdot 184](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-4ef0bd49fbc679527c97f049857957b0_l3.png)
Just say 184 over 2 times 10; therefore,
.
![Rendered by QuickLaTeX.com \bold{920}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-76d79c4c33b9913c3b139e0947047b23_l3.png)
What is
?
![Rendered by QuickLaTeX.com 5 \cdot 121](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-772928841f2ca56664140d56ecb05ef6_l3.png)
Just say 121 times 5; therefore,
.
![Rendered by QuickLaTeX.com \bold{605}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-448010ea16b46e0a483fe6be62d8cf21_l3.png)
What is
?
![Rendered by QuickLaTeX.com 3243 / 5](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-9344343560df4a7b32bb0a1586b3839a_l3.png)
Just say 3243 times 2 over 10; therefore,
.
![Rendered by QuickLaTeX.com \bold{648.6}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-86a7368721c89c819df2182f04b44498_l3.png)
What is
?
![Rendered by QuickLaTeX.com 2 \cdot 4585](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-c8485bdbc1e44099eaa5b273a7c6805b_l3.png)
Just say 4585 over 5 times 10; therefore,
.
![Rendered by QuickLaTeX.com \bold{9170}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-0e769e64a88d8ae6b7fca3e53c3619bb_l3.png)
What is
?
![Rendered by QuickLaTeX.com 2 \cdot 1236](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-dacd4b14028aa75ff970100c0e2bb9bf_l3.png)
Just say 1236 times 2; therefore,
.
![Rendered by QuickLaTeX.com \bold{2472}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-1080e560e90a6c79ff8e6d7729148a95_l3.png)
2, 50
![Rendered by QuickLaTeX.com 50 \cdot 2=100](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-0c4de330ab30fcc81e69db97bf472ad5_l3.png)
![Rendered by QuickLaTeX.com 2 \cdot 50=100](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-27ce9f6b72bc85205eab768d3f46ea1b_l3.png)
5, 20
4
![Rendered by QuickLaTeX.com 25 \cdot 4=100](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-a9729be59ccd983c6606b05b4f51aeac_l3.png)
![Rendered by QuickLaTeX.com 4 \cdot 25=100](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-b57699cd51ebf1d28f5b2a1b5f6b835f_l3.png)
To multiply by 25, we can divide by 4 if it’s easier that way
What is
?
![Rendered by QuickLaTeX.com 25 \cdot 160](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-b4705e5c8272ec923e98cc229204aa4d_l3.png)
Just say 160 over 4 times 100;
therefore,
.
therefore,
![Rendered by QuickLaTeX.com \bold{4000}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-103a18591b9d08ada78fd25b3e272d8a_l3.png)
What is
?
![Rendered by QuickLaTeX.com 600 / 25](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-9ab2118dfe570dd7350abe0f8ecd5839_l3.png)
Just say 6 times 4;
therefore,
.
therefore,
![Rendered by QuickLaTeX.com \bold{24}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-9edb664ee92453359c555d7320d48442_l3.png)
8
![Rendered by QuickLaTeX.com 12.5 \cdot 8=100](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-4c8a5a0c9e607c8e545efddc19b024b6_l3.png)
![Rendered by QuickLaTeX.com 8 \cdot 12.5=100](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-6d9c2b2276ed5ba13f5c12c099c1446e_l3.png)
3
![Rendered by QuickLaTeX.com 33.33 \cdot 3=100](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-4baf642c986e25f8d5de6ee4ec9b1f12_l3.png)
![Rendered by QuickLaTeX.com 3 \cdot 33.33=100](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-74f25c39c426168443896e8e0592853b_l3.png)
6
![Rendered by QuickLaTeX.com 16.67 \cdot 6=16\frac{2}{3} \cdot 6=100](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-9255a7fa7b29000ce42b2a8152ed4c2c_l3.png)
![Rendered by QuickLaTeX.com 6 \cdot 16.67=6 \cdot 16\frac{2}{3}=100](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-091923c3d1cdbaa869e0291af680ebec_l3.png)
7
![Rendered by QuickLaTeX.com 14.29 \cdot 7=14\frac{2}{7} \cdot 7=100](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-e3d1a58a942a890de021f7ad50ba8b9a_l3.png)
![Rendered by QuickLaTeX.com 7 \cdot 14.29=12 \cdot 7\frac{2}{7}=100](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-ea116facba32d3dec26975843e184a06_l3.png)
9
![Rendered by QuickLaTeX.com 11.11 \cdot 9=11\frac{1}{9} \cdot 9=100](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-1ea3b288750a22be51aa461e0e377988_l3.png)
![Rendered by QuickLaTeX.com 9 \cdot 11.11=9 \cdot 11\frac{1}{9}=100](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-33e93213482c19d247989720790213d8_l3.png)
12
![Rendered by QuickLaTeX.com 8.33 \cdot 12=8\frac{1}{3} \cdot 12=100](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-afbde0cb94ab19005e3ecc99f55a5b4b_l3.png)
![Rendered by QuickLaTeX.com 12 \cdot 8.33=12 \cdot 8\frac{1}{3}=100](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-777e1695ff3b65d5b5400e14a3416234_l3.png)
What is
tax on
?
![Rendered by QuickLaTeX.com \bold{\$3.88}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-51ff401b5d12d3c61d095778d1bd2325_l3.png)
![Rendered by QuickLaTeX.com 8.25\%](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-86afd69a289a79753c3edbc162a8444e_l3.png)
![Rendered by QuickLaTeX.com \$47.00](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-3e5f40760e125d155d8886fab5d9b818_l3.png)
Just say 47 over 12; therefore, ~![Rendered by QuickLaTeX.com \bold{\$4.00}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-f2f0d50806b57645338820e3bbf553c9_l3.png)
Could say 47 over 12 less say ~
;
therefore, ~
.
The exact value is ![Rendered by QuickLaTeX.com \bold{\$4.00}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-f2f0d50806b57645338820e3bbf553c9_l3.png)
Could say 47 over 12 less say ~
![Rendered by QuickLaTeX.com \$0.10](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-487c381e3ee9528744bd3a7d4e4afe67_l3.png)
therefore, ~
![Rendered by QuickLaTeX.com \bold{\$3.90}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-f9653e58d2bdaca9e4415836194a50fb_l3.png)
![Rendered by QuickLaTeX.com \bold{\$3.88}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-51ff401b5d12d3c61d095778d1bd2325_l3.png)
15
![Rendered by QuickLaTeX.com 6.67 \cdot 15=6\frac{2}{3} \cdot 15=100](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-4a1a27b57d30641be4ebee1e7229bb2d_l3.png)
![Rendered by QuickLaTeX.com 15 \cdot 6.67=15 \cdot 6\frac{2}{3}=100](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-b6a77ca9f8ee0bdb17b8d1911ee269d4_l3.png)
What is
of
?
![Rendered by QuickLaTeX.com \frac{2}{3}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-6e89c4f69688b0dd6ab75a55841059de_l3.png)
![Rendered by QuickLaTeX.com 630](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-6b5bc7fca0cbf82878f989a7ebf57365_l3.png)
Just say 630 over 15 times 10; therefore,
times
; therefore,
.
![Rendered by QuickLaTeX.com 42](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-663d228316bfd115aace82901fc82ec6_l3.png)
![Rendered by QuickLaTeX.com 10](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-5450913cc453faf132dc56e5965ca797_l3.png)
![Rendered by QuickLaTeX.com \bold{420}](https://www.mathpax.com/wp-content/ql-cache/quicklatex.com-62a9b5ab91d1ce89a929ab4b8b630907_l3.png)
Click for a couple three quick examples using aliquot parts