The sum of two numbers is 37. Their product is 312. What are the numbers?

Question

HIX Tutor · Accepted Answer

undefined To find the numbers:
Step 1: Let's call the numbers x and y.
Step 2: Write two equations based on the information:
x + y = 37 (equation 1)
xy = 312 (equation 2)
Step 3: Solve equation 1 for x: x = 37 - y
Step 4: Substitute x in equation 2: (37 - y)y = 312
Step 5: Simplify the equation: 37y - y^2 = 312
Step 6: Rearrange the equation: y^2 - 37y + 312 = 0
Step 7: Factor the quadratic equation: (y - 24)(y - 13) = 0
Step 8: Solve for y: y = 24 or y = 13
Step 9: Find the corresponding x values:
If y = 24, x = 37 - 24 = 13
If y = 13, x = 37 - 13 = 24
Step 10: The numbers are 13 and 24.

Eva Adamson · Answer

The two numbers are : 13 and 24

Let #x and y , (x < y )# be the two numbers,such that sum =#x+y=37=>y=37-xto (1)# and product # x*y=312...to(2)# Subst. #y=37-x # into #(2)# #:.x(37-x)=312# #:.37x-x^2=312# #:.x^2-37x+312=0# Now , #(-24)+(-13)=-37 and (-24)xx(-13)=312# #:.x^2-24x-13x+312=0# #:.x(x-24)-13(x-24)=0# #:.(x-24)(x-13)=0# #:.x-24=0 or x-13=0# #:.x=24# #or x=13# So, from #(1)# #y=13 or y=24# Hence the two numbers are : 13 and 24

Eva Adamson · Answer

#x = 13, y = 24 and x = 24, y = 13#

Let the numbers be represented by #x and y#

The sum of two numbers is #37#

#x + y = 37#

Their product is #312#

#x xx y = 312#

#xy = 312#

Solving simultaneously;

#x + y = 37 - - - eqn1#

#xy = 312 - - - eqn2#

From #eqn2#

#xy = 312#

Making #x# the subject formula;

#(xy)/y = 312/y#

#(xcancely)/cancely = 312/y#

#x = 312/y - - - eqn3#

Substitute #eqn3# into #eqn1#

#x + y = 37#

#(312/y) + y = 37#

Multiply through by #y#

#y(312/y) + y(y) = y(37)#

#cancely(312/cancely) + y^2 = 37y#

#312 + y^2 = 37y#

#y^2 - 37y + 312 = 0#

Solving the Quadratic Equation..

#y^2 - 37y + 312 = 0#

Using Factorization Method

The factors are, #-13 and -24#

# - 37y = -13y - 24y#

#312 = -13 xx - 24#

Therefore;

#y^2 - 13y - 24y + 312 = 0#

By Grouping;

#(y^2 - 13y) (- 24y + 312) = 0#

Factorizing;

#y(y - 13) -24 (y - 13) = 0#

#(y - 13) (y - 24) = 0#

#y - 13 = 0 or y - 24 = 0#

#y = 13 or y = 24#

Substituting the values of #y# into #eqn3#

#x = 312/y#

When, #y = 13#

#x = 312/13#

#x = 24#

Similarly when, #y = 24#

#x = 312/24#

#x = 13#

Hence;

#x = 13, y = 24 and x = 24, y = 13#

Hazel Anglin · Answer

undefined The two numbers are 13 and 24.