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mhanifa

Answer:

  • 15 and 16

Step-by-step explanation:

Let the two numbers be x and y.

Their sum is 31:

  • x + y = 31

2/3 of one of the numbers is equal to 5/3 of the other number:

  • (2/3)x = (5/8)y  ⇒ 16x = 15y or x = (15/16)y

Substitute this into first equation and solve for y:

  • (15/16)y + y = 31
  • (31/16)y = 31
  • y = 31 ÷ 31/16
  • y = 31 × 16/31
  • y = 16

Find the value of x:

  • x = 15/16 × 16
  • x = 15

The numbers are 15 and 16

igoroleshko156

Hi Student!

The first step that we must take to solve this problem is to extract the important information.  Looking at the problem statement, we are told that the sum of two numbers is equal to [tex]31[/tex].  We are also told that [tex]\frac{2}{3}[/tex] of one of the numbers is equal to [tex]\frac{5}{8}[/tex] of the other number.

Now that we have completed the first step of information gathering, we can move onto the next step which is to create a system of equations that captures the information that was provided.  First of all, we know that we are going to have two numbers being added together to [tex]31[/tex].  So let's start off with that and create the first expression.

First Expression

  • [tex]a + b = 31[/tex]

The first expression just captures that we have two numbers that add up to 31.  However, we also know that [tex]\frac{2}{3}[/tex] of one of the numbers is equal to [tex]\frac{5}{8}[/tex] of the other number.  Therefore, we can create a second expression that will capture that information.

Second Expression

  • [tex]\frac{2}{3}a=\frac{5}{8}b[/tex]

Now that we have a system of equations that incapsulates the all of the information given, we can move onto solving the expressions.

The first step that we would want to do is to solve for one of the variables in the second expression.  In the example, we will solve for a by just multiplying both sides by [tex]\frac{3}{2}[/tex].

Combine like terms

  • [tex]\frac{2}{3}a=\frac{5}{8}b[/tex]
  • [tex]\frac{2}{3}a*\frac{3}{2}=\frac{5}{8}b*\frac{3}{2}[/tex]
  • [tex]a=\frac{5*3}{8*2}b[/tex]
  • [tex]a=\frac{15}{16}b[/tex]

We have now found the value of a with respect to b which we will now use in the first expression by plugging in [tex]\frac{15}{16}b[/tex] for the variable a.  After plugging in the values, solve the expression.

Plug in the values

  • [tex]a + b = 31[/tex]
  • [tex](\frac{15}{16}b) + b = 31[/tex]

Combine like terms

  • [tex]\frac{15}{16}b + \frac{16}{16}b = 31[/tex]
  • [tex]\frac{15+16}{16}b= 31[/tex]
  • [tex]\frac{31}{16}b= 31[/tex]

Multiply both sides by 16/31

  • [tex]\frac{31}{16}b= 31[/tex]
  • [tex]\frac{31}{16}b*\frac{16}{31}= 31*\frac{16}{31}[/tex]
  • [tex]b= \frac{31*16}{31}[/tex]
  • [tex]b= 16[/tex]

We were able to solve for b and retrieve that the value that it stores is 16.  We can now plug in the value into one of the equations and solve for a which will give us the value for the final unknown.

Plug in the values

  • [tex]a + b = 31[/tex]
  • [tex]a + 16 = 31[/tex]

Subtract 16 from both sides

  • [tex]a + 16-16 = 31-16[/tex]
  • [tex]a = 15[/tex]

Therefore, after using a system of equations, we were able to determine that the two numbers that were being referenced in the question were actually 15 and 16.

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