A grocer sells boxes of six apples
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiiblYO40VRNL6tyqud8Ppr_yq-A-VisodJ-l2-4m_H5-raK44IvoIYy8gccUa51RwtybjZyf8qwztHnMCLzQDXaCFxlYvOHV5GDku7Vv26C3MpjhUVwZeqKjhnyfhhrsYa14y41NnVo_VU/s1600/image317.jpg)
A customer orders five of these boxes to be delivered in a box as a gift.
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjBueSAm-H_ODP688uBb7z5jPbz6wn3PHD7U3m7fwsTnZCCBw-62uGZJux4y0CmkCWcUEzt5AwhPlfHdDuawa3GBKV9ojTq2l8sMgJ5gmrlt4hm3LQzW-EqYUSqCAKwZsHxtyAiHmGA-b73/s1600/image318.jpg)
How many apples are delivered in the gift box?
Five boxes of six apples, so 5 x 6 apples, which is 30 apples.
A restaurant seeing this gift box orders four of them, how many apples will the restaurant receive?
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjBueSAm-H_ODP688uBb7z5jPbz6wn3PHD7U3m7fwsTnZCCBw-62uGZJux4y0CmkCWcUEzt5AwhPlfHdDuawa3GBKV9ojTq2l8sMgJ5gmrlt4hm3LQzW-EqYUSqCAKwZsHxtyAiHmGA-b73/s1600/image318.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjBueSAm-H_ODP688uBb7z5jPbz6wn3PHD7U3m7fwsTnZCCBw-62uGZJux4y0CmkCWcUEzt5AwhPlfHdDuawa3GBKV9ojTq2l8sMgJ5gmrlt4hm3LQzW-EqYUSqCAKwZsHxtyAiHmGA-b73/s1600/image318.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjBueSAm-H_ODP688uBb7z5jPbz6wn3PHD7U3m7fwsTnZCCBw-62uGZJux4y0CmkCWcUEzt5AwhPlfHdDuawa3GBKV9ojTq2l8sMgJ5gmrlt4hm3LQzW-EqYUSqCAKwZsHxtyAiHmGA-b73/s1600/image318.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjBueSAm-H_ODP688uBb7z5jPbz6wn3PHD7U3m7fwsTnZCCBw-62uGZJux4y0CmkCWcUEzt5AwhPlfHdDuawa3GBKV9ojTq2l8sMgJ5gmrlt4hm3LQzW-EqYUSqCAKwZsHxtyAiHmGA-b73/s1600/image318.jpg)
Four gift boxes, each containing five boxes of six apples.
Four gift boxes, each containing 5 x 6 = 30 apples.
Four gift boxes of 30 apples, so 4 x 30 = 120 apples.
Using () to show calculations in a box we have worked out 4 x (5 x 6) = 4 x 30 = 120.
We could have done this differently.
Each gift box will have five boxes in it
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjV1jLf0zsAcFcoXdGwnFC0ncQqaMwsj2KXjIMOhdDtqSH2VIhDOsJd_OqKtsWPe5q7VFgBHOJiKf7Df6b21Ddp2qw8-t_KhaKNMhQsNDQlGmYf4U7HwPIiF9Q4RVnnlcquPnqfhQS7ARrA/s1600/image323.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjV1jLf0zsAcFcoXdGwnFC0ncQqaMwsj2KXjIMOhdDtqSH2VIhDOsJd_OqKtsWPe5q7VFgBHOJiKf7Df6b21Ddp2qw8-t_KhaKNMhQsNDQlGmYf4U7HwPIiF9Q4RVnnlcquPnqfhQS7ARrA/s1600/image323.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjV1jLf0zsAcFcoXdGwnFC0ncQqaMwsj2KXjIMOhdDtqSH2VIhDOsJd_OqKtsWPe5q7VFgBHOJiKf7Df6b21Ddp2qw8-t_KhaKNMhQsNDQlGmYf4U7HwPIiF9Q4RVnnlcquPnqfhQS7ARrA/s1600/image323.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjV1jLf0zsAcFcoXdGwnFC0ncQqaMwsj2KXjIMOhdDtqSH2VIhDOsJd_OqKtsWPe5q7VFgBHOJiKf7Df6b21Ddp2qw8-t_KhaKNMhQsNDQlGmYf4U7HwPIiF9Q4RVnnlcquPnqfhQS7ARrA/s1600/image323.jpg)
So there will be four gift boxes each containing five boxes, so 4 x 5 = 20 boxes.
Each of these twenty boxes will contain six apples, so 20 x 6 = 120 apples.
We have worked out (4 x 5) x 6 =120.
We can only multiply two numbers at a time however we have found out that which pair to start with does not matter.
4 x (5 x 6) = (4 x 5) x 6 = 4 x 5 x 6 =120
As it is with boxes of apples so it is with cards and holders.
Consider this hand
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjqXx_VuQRQITOpbM3-DDHkMbdEQyiM48dxvPodb1kGKJZc2J9SNxsz2trok39ygkigxvSKAtBpna7295Lp7015Dxyhyphenhyphen6my-NGdincuKbOOJ21E9N4YZ6ZDQ-CFvK9iYx_yUcSDFVMjj8mI/s1600/image325.png)
Using a new holder to hold these four piles we get
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhrguGcMdL1Ot6yVqljw20UfAe5MMnfy_U9r0QX_sD_nh16xKwko_ymjm_Yt_rzWIVoRzcFauvT9w6q70EolRJ04k16yn6ow6DAADbXETGsbehxDLl5x02YY5gid_G9tYk3dEfBjE4EHuj8/s1600/image327.png)
and as
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgtVrtrPcDlPO5bACX_dtkvACKHm6mE6tbPfqz6AME9GbBbpl_AaYVvwy9nSEBz78vgkbr4nlU8HNpl5re9-FOROsowY4WiUrN3OoQ_OqF_8Cj3RLuzyIzUTMMai4aa5bJ5tRI-Hkkd6lvO/s1600/image329.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi3x7DMcPAfGBB_v8xTOe-kysJsAXithWpDn25NtLR0U-XcgCJ2iqVpZeYFc68oQ4P3YYbqXt6hCZk_5T8N_udnQj6bXIMCG7YErHUEBhN2oHDJwoVMWMA-8IWyTiXkvMzXtS1diCh6TbxR/s1600/image331.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj37FhqxuCVL8H1hGAsGjgcsgPH0g5o-pdvV8O6T_kUUCXrYtfXh1UJSKThORmPN0sXJxBDKgeW3vx2Qh8Gd_HLnlaa13Ju1AmB9EEsWFhyphenhyphenadJAscbUVftn1iaK_SxX9mLkF0iy3_6QSB7d/s1600/image333.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhJQMz2sYr1T03D12CnqQZG9SKuW8YSdkXWRuklZfCvekyxuNdfHWuJ-wXAECFrt_Ltd2YNcw-clh-Ae9FqXRB0igv3RdfMbUAJeu567jgNWfMlnAHaTAja5DNKFbYRc_vlk0luZwcHSv9R/s1600/image335.png)
We could have done this differently
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgk_WWeyASknni4aB0x1tBIcC0pizzdtDwrjxX-JHP1hqJ5CMvzclFbq18LrKcI8jEvZezshXdhxjUoMrm4sxeLvKMLIw9aZlgQUo0FCgqfIXzl-eLsVFsfH7-wj9c_H8qurOIOzZMNvrEW/s1600/image337.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjuoo8CHQvsUdfcEK5KGP9kRmgw8kUGpiBu8PP_d7lRY12UepFtCWIfUPEfY2eXG-OduM5X72OrJc0TAwE3vqYTpF57XPVRNvimyj5oZwXJ8A1MpBsMAoIYxflX_noRjcqL2hQUD_VLCfuo/s1600/image339.png)
So
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhM1GsN9Ctg2DhlpPiHvGxs1Kiqt5L_rNy0x9P6VV5IZkw4ftqm20nd1bLGNg3hpyXu5d90LXm-Nmqrk5UtqD7iS2GMZ5Y4dJ2fYMFpIG92Vpjc-bLR9M-XZR5XOwWQc8BNuFRQ52haR3iI/s1600/image344.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjBGxOuyDTY7yCXim_fPxO1G4q0rufAO_OALPxt2U2HNXRq5Hastdq860UlYloH18XFTrErnksUMuu8oR0BXtzBMRZ5DVal6h1Hvmy3odJClVRpu3ZkUE5E6w1T2fY3a8x5JtY1bxpZ7UZj/s1600/image342.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhM1GsN9Ctg2DhlpPiHvGxs1Kiqt5L_rNy0x9P6VV5IZkw4ftqm20nd1bLGNg3hpyXu5d90LXm-Nmqrk5UtqD7iS2GMZ5Y4dJ2fYMFpIG92Vpjc-bLR9M-XZR5XOwWQc8BNuFRQ52haR3iI/s1600/image344.png)
When we have four gift boxes containing five boxes we have a total of twenty boxes
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjYegPoGsChZ43NqH2l-GrledFZFNeF5V1WE7W_wI0MbZ_jildBoJR16ekSV7-6ujZWBhqqpBDej6OvBNhRKguUL3qjmisXWwoUveqMHlKZKHt4-ypKaP3_hU7rNyN1g-n9IgNh7n1RbuUA/s1600/image345.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhi73O3DcBQrs4qT_EUhxsQM0mlkP0VuAAMqHi5W7_AwpxcJcdY-L_AYVqlP9tyuiGSbzxUOiTMNahM85CiQSY6d_PUsR0wQrHzAbMbh4O9QLtoVd98CoI6VcZ-DVrxopEWZrF8kdEoR-QY/s1600/image346.png)
These can be re-arranged as five gift boxes containing four boxes.
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEinc4800AcOx40ggm4H2XaDUPKXAYDTvSfCOovf9lAPJihr175Mwfydg9RVIuI55zzECMxv4rEaQF_hY9FbQv6rfRn3vWCy_k8VV0ko2fvPA7_ok6UJmfOh0cXqBIs-86lmlUGxnPMc8f9w/s1600/image347.png)
In this case there is still a total of twenty boxes
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjYegPoGsChZ43NqH2l-GrledFZFNeF5V1WE7W_wI0MbZ_jildBoJR16ekSV7-6ujZWBhqqpBDej6OvBNhRKguUL3qjmisXWwoUveqMHlKZKHt4-ypKaP3_hU7rNyN1g-n9IgNh7n1RbuUA/s1600/image345.jpg)
We have already seen that
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9apoBDwIQIpvlUtgbTILBpsbhTWao4-CMJvG9G8uoSKQI4IReCSjXqvyIkMad-ntIInLV08aN7UAxYab_u5xmoZ7NC90b6Wm5whqL0feS8O52uOOg5JdDSFMGrCBOzA57Vsl_nKTqj7Qy/s1600/image349.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhajBA_dApWxkAsjujztKXxWNRWwctho8Id99ttZNTCbUvv2fyTylacRarhKYMbGXsIZTlYd_1U582EDrZkV6BGQw8GS3yxk6_7kWG5J4K9V2PuZlI_7H0ktr6cfRX8Ipe-hz_4Vz63dNXA/s1600/image351.png)
Thinking about the boxes above it is also true that four holders holding five holders
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg2fhqZ6qsFj0dKRFGd88ixZNzlqR8ZSzviXyqrYfly9hW1mt6yzg6Lzm-fQ_B_b9R_6gPYKZ0IMW1y-yYsR8Mojt1F8SKXmyMiKoGoHckhBWYNyzMnDCiq_aZSSZC3KGy1pIFh-wKT-6rO/s1600/image353.png)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbHYDzlUO2MFvCGt73QJalCVhshrAo4H0TCTEKKrUKeo_yNIG62y8zGshKMLHMTgcn1-3QGgl_KZQ3ApYq4rMtDuJQNK9WbGc2mvSgAetsqh2okKko2i1SJfLRVWx0ZII3sEU-Hn2ZdYIz/s1600/image355.png)
All of the following give the same result.
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiSMGWARg3l2YiOZmW_qo_Akvw_pcJdEYKE_BqGb2DZdl8u_BWPcTcor4rAsSaTldtQB_O60u9Jm_hIBrUGCIAavFKLiLEoiZRp1A3l22Jy4z47uX2k78HjlefpAgt2_HGONsZH_UzjN6ee/s1600/image357.png)
Recorded as 5 x 4 x 6 = 4 x 5 x 6 = 4 x 6 x 5 = 6 x 4 x 5 = 6 x 5 x 4 = 5 x 6 x 4 = 120
When multiplying numbers the numbers may be multiplied in any order.
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