![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgF7f7jaK_ir1gPLf6HQaumSnM-fcb3I60sVcAWWDPte1sblyOTLkDAc2eYFjXurQ5SHW8E5jVkJIqvPWSik7ZL56VnXRw7WdPVwQEgxwyH7RFt5_SJkx_0Us3lxImMhpj3xRJ9SEwAVuEB/s1600/image002.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEirdCoBHIGrY2tkmQB0Rb1XOYOMNL5HynY0bjfNlUj7jRyKF9vx1yt0zL1vGRRX9A89JZFx70AAAdAEW6dTpetUta2n-We_nOdoLaygr7BEkPgmgokva2uTr4yMuK7i-ApgVGQmDDNNDXw8/s1600/image004.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjZPFB5VT-oz07WWb15uGpfwJUSpiLrERMXPMuGysGidbeNEDGJPDqqxZdr0A2NE12BczcN5SjqL-6BjwaZF43yi4oA8uroOScjIkmkqtvIE4VwDKcR-cffGHFFhWnV3HKPputcr7Vd0jZ4/s1600/image006.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjA4w20-a37gqTMANTC3TP41USBX5LArlfpI5C0OeeDbyQ2vdotHq0wUyLWNeOAonF0skyMukTjv1LjsZQcq0A6Oq6s7QIjRlT3u4XmIZr3ifThmFj8weLFAMm2VFyFkH1qyn_snHNyI8CE/s1600/image008.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjampgytO31F1H_n1gmUPlHlPkTq679aHe8zmqzwqu07M29-p1djZ7RoCCDcB7PzkucNhmDnIu7BGlks8EedTCXt5vBfJhEyHMpNXVBSq7DQ-cbK3zwFUPFHoGt8I_4topx7dwRTIWb7z8F/s1600/image010.png)
More generally so do these hands
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgNMGpj1iRlApQ0IPkK82mvs9LoKFxRly347kgsU_rGqYXL0yHZttWiUeFleJcBazR47d1DaGHVh3GaB4cNJrW41r_y1YrXSM0fLM5pHDLQZHELJP5k2LpOfY3A62fyRn8NRBoBkS1wVpgJ/s1600/image012.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEipPBuK8fOrIedxMeDBsMAqF1wRJGWwFXOu63fUEmYHG3L2GSWyCXxBteHlFID6LjMUOI1YUqe9WS684fqJdxs0oNvpkrMgR51crBGgaHHNrMMyKeS1l4KN1L1h9fHcv2YHAazuzF96tv0o/s1600/image014.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg4K1MtHNadzA6DVMo6njnakNvIgpVHI_ltKYNdX77X6Pv2B6zz7YdaP_-u-8xVkIuf99lIw4Anv2SE2Ad8rKM_oOT3eTdsPj4Vuu1ncmvtdn9z_Y_8kKALiVr_sepEgbEM24RvwFLOJHNP/s1600/image016.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhdNoW7HAD_QQ4WXo6b0k66GeskbRo42tJyJsY8l7c7YT_havUO_WcZ8E_2qhOD3p-kaRwIm3sriUo-a8YmVd5mXn_naZUk9rghG-Z0ecjNbLa7Xwmj2Q-gVcAMDpfQqTvqOKnY-dZWGo9v/s1600/image018.png)
Whatever the number x is declared as
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgWXHXi9K139bDx4u200yCfYXih_7x0RlEZ82D261H7-2IV70vy90dhoE9z6IRT1mPqrohUAeYSIH3YjG0Qw49iM5_LEwB6mPWAdFTY3A2WHFTn6jb1hKWpTPJ6orlq4Zv4skzSqTuT6SeL/s1600/image020.png)
We have already seen that the solution to
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjpf4m34z744AMqEyPnsc8noh9-XzQSHiw_0Rz8yexYYNpflCtmlCqM79Ejmw7-J35lRWJ1zykJmbTEYZcqdjtdvJ14ofC0xGUXJ8iaEG1HL0ctDLoEX4b636aX43erLesBoA7IXQirA0pp/s1600/image022.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjPMLad_urlfTjZYpYH3mVACnUOwTwxLc5L3pYVgpjN1LO3CmZPr1j9_XvZNE8SBMkn5YFfDN__7q_XBVkmRgEQlVXXsp79gdMuJ9ktQVAPpYE3JjJH2VaM4VgHpacA93x1qi2rVGuescmS/s1600/image024.png)
How about
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj4SlFkeltZe6zpDoXX-tjP1qKeS3Y1YHib7ibmggzL-d8VeJ-O3xMKjacCCIlGlZh9o3JeThtAbbc-Rg2YkSMsiFPY2Q6kKYGuRTVWvm5VdtadWkokhq4jKjxZwESeKKeJWpijyeEsq5SH/s1600/image026.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjtiA2FEgEg2XDdABRvPoqWhGwk5XGhyvxUAchYJPZWAAQ-0F9oARnWFDSx6IqxKAaZOUIDI04OM3J34dkcIlc2bNshnoN_1cRBppVN6QlAGTL3TvrGdCE1JQi9FlauuYlR8d3AO-EEWDVr/s1600/image028.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiuskEOEAje5WRqV2-BHW5-YdgSEFq3cCyEDeetLfrfs81qnSzCTWn8J9UqyZXHX1F4URL4mlLS-7lwB63JqJGPyUPSYkziAidgPqTTtzHgxKlY1pu3bccVKgCb4GYvXnMY2_kXWNzwddaO/s1600/image030.png)
The rules for solving
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj4SlFkeltZe6zpDoXX-tjP1qKeS3Y1YHib7ibmggzL-d8VeJ-O3xMKjacCCIlGlZh9o3JeThtAbbc-Rg2YkSMsiFPY2Q6kKYGuRTVWvm5VdtadWkokhq4jKjxZwESeKKeJWpijyeEsq5SH/s1600/image026.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhjDRI4kk-zHWsU96v71Ap-YlSWMFnIU6rFj5okalOgEoqJqo0nGp9W-fCrmN7iKCcGgGBQW8vtc2f_CJPd-T3IWCu5S6XFkHKfq7gkAL3ANRxSJ-Mcr7CeKedRYvmQUL_6QJeqeZWN-PLn/s1600/image032.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjheczrE3KteX9CfvzOMOnZCtz780Y8HPbugEdjSLQLQNtzR8N7w39M_MFvobqmEiyxKjZ9bG62s71ETxl_rSm5hc8dmgfxlg8TxK9MCSaHpuLtfBovNtw5gV4nU4gKfUgbAI7QnrOFOgXj/s1600/image034.png)
This is true for any number x
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjGLPpK2Tp9PxW_AvclRx_C2X3EozMYiEuxrdKHs-rHxFYRPXUHPKpEv98z5nwfgiyHDR20NMtv3TDWQ6xWjC4CVowwU_cfrjenZ2n-aOWV1eiszLxeTUcbR6iZ64tsKhDyTnOEmGjlJk0n/s1600/image038.png)
What about something like
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgGvrQnWChq4PNN1UuxtucYtk41co-pCuDkTEYLiWA-tnQoBHRtkZazyToXraaDDBy39j355oZcXQlenkbO4xKdzDAtBNpA-G8AV97f6kY33jOzXnIQ368ELXPe_lwuVyol3i3JYxVAtUZb/s1600/image040.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgIuSKcJX8wowCDv-SY5OM1MVkbtIxxX6JF8P9dJjW8p_n20L_PumxUfsemzCCcPH4y_qJtdBX0JRhrSSs3AlKa7xibGj1_2bFsyBu7M7VgGHUt23C_JS5GpqS66F5fjKbMyydr0F6C55u9/s1600/image042.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjKHCzZacVhInaJXRPR-5cqtgl5Bq54lqn8lW7j1NDGxJ0_YuBjbNZuT8_ULTTZ1Om5bsvF4KOIyOcOWPvkZYpilWasb8uffFHwqJthhwkrHW8WNlMkHWQ8Xa8y8BZvq_6OBf3NOI-kFw1r/s1600/image044.png)
When
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgIqvBKiJllE5b35_8skN99h7cFr8OLMCFJY6lm6h9sYz936QB-qfgfMNaIswaq1E0wG6Z4OhweUUFXUueWTWQbkOzR8uKKi6NlXaDEbRywnDq36RK5Yh9XcMNsL_s-jFAfl_NV_KvNTo76/s1600/image046.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhw-6ayyNmf7wbFv2M2c3DBTvcsbwzuUPHQ-OPn9aWxMWvpHbbueSQhB_YT8iIXRC7vo6kcUnVj73WB1QaGJvyOuJfsmuP3bneYk5_q3y0fa036j-Z9okwATSVSpUNBj7iPzw5ewN5FwuAe/s1600/image048.png)
When
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgARg4HX8y7RFVWBVF3RECr70nSMBc8dRYzSuov3mK_bN01lLWvTHjuPKfABS3jI2QCjJNd1G19jKrCcmtB4ogzvLi8jHHw1PsOCukqgp7O-zuzV3Js8Xan3EFziwO-cCx-kncGr24m_lX9/s1600/image050.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiK18GFzuv1Qxk2MWAKm2vDLUBiUmxqrqAa_veTK6gYpkuaxyNBRTNhO6kSCqsbT0u51xK2vhzfj_PMThTbbGE6rCLXCwNkjB02HKSQuxEPPHrXHikIUCkX5zFDDH643ZxFecqNh6F7vXiy/s1600/image052.png)
The solution to
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgIuSKcJX8wowCDv-SY5OM1MVkbtIxxX6JF8P9dJjW8p_n20L_PumxUfsemzCCcPH4y_qJtdBX0JRhrSSs3AlKa7xibGj1_2bFsyBu7M7VgGHUt23C_JS5GpqS66F5fjKbMyydr0F6C55u9/s1600/image042.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjI-R-PGIzX4-Fo-hSQmofmUc0Q7K_IGrQpm8JLOnnG9K77luOMRCpN-M8fJ0APXd8KSuYWNLd9pseV-VIu35fP-kMvlte96IpnxKCu8JOtD_6L0NMRn4w48gEPUxdsY6jwBzhwf0HJz06L/s1600/image054.png)
A new name when One is divided by a number the you get the reciprocal of the number
The reciprocal of 6 is
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjNdlQA34IYMedeaXFIawRGlxyCQiC_5RWh2aT1bqpmtT26dBNrJXU16SWHn3mglKlLUONamyoCPHM9ArHqRSL5-Zs5b2JxQ54C6wX1KOtua4W93fI_bW6dh7Qab9__QBRpmKYwlZB-0qI3/s1600/image056.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhg7eLNrFZlvuCFedCnNUmPN_rzulJQ7wdxzEOvrEGThkftuXKe_6K_slkzaJFBTXLfhMj4D7lC2SvKsLlJ0v6X8j6Uq-gjGhp2birNO9Y_3UBg6q3DwnIoeJdcdUNwhks4OjipfWLbN-_H/s1600/image058.png)
The reciprocal of
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjNdlQA34IYMedeaXFIawRGlxyCQiC_5RWh2aT1bqpmtT26dBNrJXU16SWHn3mglKlLUONamyoCPHM9ArHqRSL5-Zs5b2JxQ54C6wX1KOtua4W93fI_bW6dh7Qab9__QBRpmKYwlZB-0qI3/s1600/image056.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh-zrGXlChuD6K0uVFocTC6Ze_rlTxHCCIASAv-AkaNd3SCU41y_7al3q-axTMtwPcF-0MPHPZUKB1maqN5laBtjTEsFivCfMbCh01woP7w8JZABSS3DZ5lC-5y49Vzjbd3Ae02YQ0qHyDV/s1600/image060.png)
The reciprocal of
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj4CNQJDJjetDluBJIk9LQm7DsQ20ntUjcmldDdN53_n0-6zllEDbc18x4XmC4uIiuAQHLg1CDrxYdpbJ8mme8Ci2NIWzL3DUwuhWLRZP_MSbUDY5otZQM_K3_hYz-ogikdLj1WK5XMgR4X/s1600/image062.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgiljPnT5RRfTVmpiMWyFOTLkfunR3Uan8bs8NqTCvg-cODC75c6rKpQiAi26zLxHZBTRHjzrKK58efxaHvjyabyxz9BkW4vUScxf5xPBFfv_ifR-36je6Wp18ZaUGdBjoygVK3KSJp9dtn/s1600/image064.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj3c4UXaRembyi2idh_stWMYPucVLuPQgqhOsUAJZBGVBcsWaaCSA6Ut04eA1fVPlsysN6MUcfvjfemqR2Wf0xoAEqvSgZk_v-OrbDf-o12RedI1FEKZu-akEmufFrWZlK-8uYCMVybOB84/s1600/image066.png)
The reciprocal of
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgiljPnT5RRfTVmpiMWyFOTLkfunR3Uan8bs8NqTCvg-cODC75c6rKpQiAi26zLxHZBTRHjzrKK58efxaHvjyabyxz9BkW4vUScxf5xPBFfv_ifR-36je6Wp18ZaUGdBjoygVK3KSJp9dtn/s1600/image064.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj4CNQJDJjetDluBJIk9LQm7DsQ20ntUjcmldDdN53_n0-6zllEDbc18x4XmC4uIiuAQHLg1CDrxYdpbJ8mme8Ci2NIWzL3DUwuhWLRZP_MSbUDY5otZQM_K3_hYz-ogikdLj1WK5XMgR4X/s1600/image062.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6bjXKBG9sWaf_dxF-uAri7NxGfYiyAnBjcMmORNlbtPKqaqRRNDfqEIPJahkNDbd1jaXqU4yMHPVMPqj7vTBUOjVS81wwzN8bUDQwFSDrjNic8uR5avqKdHRxqADWn5yHfojWyo2XQw23/s1600/image068.png)
You form the reciprocal of a fraction by swapping over the numerator and denominator.
The product of a number and its reciprocal is always One.
In Dividing Fractions
We found that
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgHFVhDGPsoZhbZVplPgvB_jw0YHKwKcosQbVYqAZXFYmZ727L6dsYOuUR0YCWYNGxGRbPABu6N9j8LuQ2qhwHY5Q7z852UmSKvptn8bzozxRYm4iF6qR61-7LV-hEnWXJz8i8Hbp63EtmS/s1600/image070.png)
Instead of dividing by a fraction we multiply by the reciprocal of the fraction.
Further Example
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhXlYTPSvFIE4IlJNaTntMFJc5psWQCa1uya-VRnGcVAmgnlC5feB1THGqO-faUc5btV9WvAFXPbt2b5CDQMkd3FCN_tApUhX2-zsxr8tydMa49XYCyHBHjbPpdp0bUqVS9jSvHTbDCZKY6/s1600/image072.png)
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