If the diagram below can be represented with the addition sentence **25 + 14 = 41**,

what mathematical sentence should represent the following diagram?

**Answer**

Notice how the first diagram is arranged in groups of 8, plus some units? It represents an addition in base 8. Therefore, when we add 5 + 4 = 8 + 1, we have 1 in the units place, plus an extra rod (eight units together). All in all, we have 4 rods and 1 unit represented by number 41. We can not read it as “forty-one” since forty implies four groups of 10. It is easier if we add a subscript of 8 to show that we are working in base 8, also called the octal number system:

**25**_{8} + 14_{8} = 41_{8}

The second diagram has a total of 7 groups of eight plus 4 units out of which 2 groups of eight and 7 units are crossed over. This means we are subtracting 2 groups of eight and 7 units from the total. The difference is 4 groups of eight and 5 units. The number sentence that represents this diagram is **74**_{8} – 27_{8} = 45_{8}.

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