Chapter 9.7, Problem 24P

To determine

To solve: the problems about three-digit numbers.

Answer to Problem 24P

The 3-digit number is 780.

Explanation of Solution

Given information:

A three-digit number, which is divisible by 10, has a hundreds digit that is one less than its tens digit. The number also is 52 times the sum of its digits.

Formula used:

Digit problems are based on our decimal system of numeration.

Step1. The problem asks for the original number.

Step2. Let h = the hundreds digit of the original number.

Let t = the tens digit of the original number.

Let u = the units digit of the original number.

Step3. Use the facts of the problem to write two equations.

Step4. Solve the equations.

Calculation:

Consider ,

Let the 3-digit number be $100h+10t$ that is divisible by 10.

Let t = the tens digit of the original number.

Let 0 = the ones digit of the original number.

Let h = $t-1$ .

	Hundreds	Tens	Units	Value
Original number	$h$	$$$t$	$0$	$\begin{array}{l}=h(100)+t(10)+(0)(1)\\ =100h+10t+0\\ =100h+10t\end{array}$

According to the question:-

  $\begin{array}{l}=100h+10t=52\left(h+t\right)\\ =100(t-1)+10t=52(t-1+t)\\ =100t-100+10t=52(2t-1)\\ =110t-100=104t-52\\ =110t-104t=-52+100\\ =\frac{6t}{6}=\frac{48}{6}\\ =t=8\end{array}$

The hundreds digit is

  $\begin{array}{l}=8-1\\ =7\end{array}$

	Hundreds	Tens	Units	Value
Original number	$7$	$$$8$	$0$	$\begin{array}{l}=7(100)+(8)(10)+(0)(1)\\ =700+80+0\\ =780\end{array}$

The 3-digit number is 780.