Three times the larger of two numbers is equal to four times the smaller. The sum of the numbers is 21. How do you find the numbers?

Question

Ethan Adkins · Accepted Answer

See full process for solving this word problem below in the Explanation section:

Let us first deal with the first sentence of this word problem. Let's call the larger number #l# and the smaller number #s#. We know from the first sentence: #3l = 4s# We know from the second sentence: #l + s = 21# Let's solve this second equation for #s#: #l - l + s = 21 - l# #0 + s = 21 - l# #s = 21 - l# Now we can substitute #21 - l# for #s# in the first equation and solve for #l#: #3l = 4(21 - l)# #3l = 84 - 4l# #3l + color(red)(4l) = 84 - 4l + color(red)(4l)# #7l = 84 - 0# #7l = 84# #(7l)//color(red)(7) = 84/color(red)(7)# #(color(red)(cancel(color(back)(7)))l)/cancel(color(red)(7)) = 12# #l = 12# Next we can substitute #12# for #l# in the solution to the second equation: #s = 21 - 12# #s = 9# The larger number is 12 and the smaller number is 9

Eva Abramson · Answer

undefined Let x be the smaller number and y be the larger number.
From the given information, we have:
1. 3y = 4x (Three times the larger number is equal to four times the smaller number)
2. x + y = 21 (The sum of the numbers is 21)

From equation 1, we can express x in terms of y: x = (3/4)y

Substitute this expression for x into equation 2:
(3/4)y + y = 21

Now solve for y:
(7/4)y = 21
y = (4/7) * 21
y = 12

Substitute the value of y into equation 2 to find x:
x + 12 = 21
x = 21 - 12
x = 9

So, the smaller number is 9 and the larger number is 12.