# The sum of the digits of three digit number is 15. The unit’s digit is less than the sum of the other digits. The tens digit is the average of the other digits. How do you find the number?

Apr 26, 2016

$a = 3 \text{ ; "b=5" ; } c = 7$

#### Explanation:

Given:

$a + b + c = 15$ ...................(1)

$c < b + a$...............................(2)

$b = \frac{a + c}{2}$..............................(3)
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Consider equation (3) $\to 2 b = \left(a + c\right)$

Write equation (1) as

$\left(a + c\right) + b = 15$

By substitution this becomes

$2 b + b = 15$

$\textcolor{b l u e}{\implies b = 5}$
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Now we have:

$a + 5 + c = 15 \ldots \ldots \ldots \ldots \ldots \ldots . \left({1}_{a}\right)$

$c < 5 + a \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \left({2}_{a}\right)$

$5 = \frac{a + c}{2.} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . . \left({3}_{a}\right)$
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From ${1}_{a} \text{ } a + c = 10 \to \textcolor{g r e e n}{a = 10 - c}$

From ${2}_{a} \text{ } c \textcolor{g r e e n}{- a} < 5$

thus $c \textcolor{g r e e n}{- a} < 5 \text{ substitute for a } \to c \textcolor{g r e e n}{- \left(10 - c\right)} < 5$

$\implies 2 c < 15$

$\textcolor{b l u e}{\implies c < 7 \frac{1}{2}}$
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Suppose $c = 7$ then equation (1) becomes

color(brown)(a + b + c=15color(blue)(" " ->" " a+5+7=15)

$\textcolor{b l u e}{\text{Thus, if c=7 then a = 3}}$
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$\textcolor{b l u e}{\text{Check using eqns (1) to (3)}}$

Equation 1" "->3+5+7 = 15" "color(red)("True")

Equation 2" "->7<5+3" "color(red)(" True")

Equation 3" "->5=(3+7)/2" "color(red)(" True")
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$\textcolor{g r e e n}{\text{All determined values satisfy the given condition}}$

$a = 3 \text{ ; "b=5" ; } c = 7$