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Write-up


Title
Using Up Digits


Problem Statement
The sentence 6.5 x 4.321 = 28.0865 has each of the digits 1, 2, 3, 4, 5, and 6 exactly once on the left-hand side. The number on the right-hand side is close to but less than 30. Can you find another multiplication sentence that contains each digit 1, 2, 3, 4, 5, and 6 exactly once on the left side so that the product is even closer to 30 and yet still under 30? How close can you get? Can you find a multiplication sentence with 1, 2, 3, 4, 5, and 6 so that the product is closest to 30 and greater than 30?

Problem setup

Multiply 2 factors that use each of the digits 1 through 6 one time.  The product should be as close as possible to 30 without going over (like The Price is Right).  After determining the appropriate factors, multiply 2 factors using the same digits to equal a product as close as possible to 30 that can be greater than 30.

 

Plans to Solve/Investigate the Problem

Since 6 * 5 = 30, I know that I can not use both 6 and 5 at the same time for whole numbers if I do not want my product to be over 30.  So I will use 6 and 4 as whole numbers.  6 * 4 = only 24, so I know I will need the larger digits remaining to be in the tenths place.

When I am ready to find the factors closest to 30 that may go over, I will use 6 and 5 as whole numbers and the smaller remaining digits in the tenths place.

 

Investigation/Exploration of the Problem

I started with pencil and paper and calculator exploring the products for various factor combinations.  6.5 * 4.321 = 28.0865 was given in the original problem. 

I tried 6.3 * 4.521 to = 28.4823 which was closer. 

Next I tried 6.2 * 4.531 to = 28.0922 which was farther from 30.

I predict that if I continue to try other combinations, I will continue to reach products that are less than 28.4823.  But to double check... 5.6 * 4.321 = 24.1976,  5.3 * 4.621 = 24.4913.  This is not working, so I believe my answer of 6.3 * 4.521 is as close as I can get to 30 without going over.

 

Now for the product that can go over 30.

6.1 * 5.234 = 31.9274

6.2 * 5.134 = 31.8308 ---the smallest product

6.3 * 5.124 = 32.2812

6.4 * 5.123 = 32.7872

6.3 * 5.214 = 32.8482

Since the first 2 number sentences were the lowest, I believe my hypothesis is correct, that the whole numbers need to be larger digits and the tenths need to be smaller digits.

 

Extensions of the Problem

This is a good way to review place value and to compare and order decimals.  You could also try to start with 30 and divide by 6.1, 6.2, 6.3, 6.4, and 6.5 to get the other factor.  Also, I could try multiplying 6.xxx times 5.x and 6.xxx times 4.x, 6.xx times 5.xx, and 6.xx times 4.xx (changing the number of digits on each factor).  As I tried some of these, I discovered that 6.234 * 5.1 = 31.7934 which is smaller than my previous answer of 6.2 * 5.134.

Author & Contact
Vicki Hughes
vhughes@rockdale.k12.ga.us

 


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