The sum of two numbers is 72, and twice their difference is 24. What is the smaller of the two numbers?

Question

Nathan Axel · Accepted Answer

The smaller number is #30#

Let's call the two numbers #x# and #y#, and assume #x# is the largest.

The first sentence translates to #x+y=72#.

Since the difference is positive and #x>y#, the second equation translates to #2(x-y)=24#, which can be rewritten as #x-y=12#

From this equation we can deduce #x=y+12#

Substitute this expression for #x# in the first equation to get

#\color(red)(x)+y=72 \to \color(red)(y+12)+y=72#

This equation is rearranged as

#2y= 60#

and thus, dividing by #2#,

#y=30#.

We wouldn't need #x#, but for completeness sake: since we knew that #x=\color(red)(y)+12=color(red)(30)+12=42#

Eva Abramson · Answer

The smaller of the two numbers is 30.

In this case, there are two unknowns, so you need one equation to solve for one unknown, two equations to solve for two unknown values, and three equations to solve for three unknowns. You must first convert the provided data into algebraic equations. Let's call the unknown numbers #x# and #y#. It is stated that the two numbers add up to 72, so #x+y=72# (1) We are left with just one equation. Their difference is 24 times twice. Let's start with how they differ. #x-y# This value is 24 times twice, so #2*(x-y)=24# Divide both sides by two to make this simpler. #x-y=12# (2) Now that we have two equations, you can solve them using a variety of techniques, like substitution or elimination. I'll go with substitution. From (2) we can rearrange it to get an expression of #x# in terms of #y# #x-y=12# #rArrcolor(blue)(x=y+12)# Now, replace this phrase in (1) #color(blue)x+y=72# #rArrcolor(blue)(y+12)+y=72# Simplify this and solve for #y# #2y+12=72# #rArr2y=60# #rArry=30# Now substitute this value back in to either (1) or (2) to solve for #x# #x+y=72# #rArrx+30=72# #rArrx=72-30=42# You should verify one last time that the two numbers add up to 72 and that their difference divided by two equals 24.