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# Solved Problem: Algebra Number Problem

Number Problem

The smaller of two numbers is 9 less than the larger, and their sum is 37.
Find the numbers.

Solution:
Based on the problem statement, the following representation is derived:

Let
$\mathbf{x}$ = smaller number
$\mathbf{y}$ = bigger number
$x = y – 9$

# Another Simple Number Problem

Another Simple Number Problem
Find two numbers whose sum is 7 provided that one is 3 times the other.

Solution:
Based on the problem statement, the following representation is derived:

Let
$\mathbf{x}$ = first number
$\mathbf{y}$ = second number
$x = 3y$

Hence,

$x = 3y$       Eq. 1
$x + y = 7$    Eq. 2

Equating Eq. 1 to zero and multiplying the result with -1 and adding with Eq. 2 becomes

$(x – 3y = 0)(-1)$
$-x + 3y = 0$

# Simple Number Problem

Simple Number Problem
The sum of two numbers is 9 and their difference is 6. What are the numbers?

Solution:
This simple number problem calls for two variable representation, say x and y.

Let
$\mathbf{x}$ = first number
$\mathbf{y}$ = second number Read more »

# Work Problem

A work problem is one in which a specific job is done in a certain length of time when a uniform rate of work is assumed. For example, if it takes a man 10 hours to paint a room,    then his rate of work is $\frac{1}{10}$ of the room per hour.

To solve a work problem, we need to multiply the rate of work by the time to obtain the fractional part of work completed. Like in the example above, if the painter works for 7 hours, then the fractional part of the work completed is $\frac{7}{10}$.

Work Problem

# Uniform-Motion Problem: Unknown Rates

Peter and her daughter Liza leave home at the same time in separate cars. Peter drives to his office, a distance of 28 km, and Liza drives to school, a distance of 36 km. They arrive at their destinations at the same time. What are their average rates if Peter’s average rate is 10km/hr less than his daughter’s?

Solution:
We need to determine the average rates of Liza and Peter; therefore, we make the following representations:

$\mathbf{r:}$  the number of kilometers per hour in Liza’s average rate
$\mathbf{r - 10:}$  the number of kilometers per hour in Peter’s average rate