The larger number is 20.
Let's denote the larger number as x and the smaller number as y .
According to the given information, one number is 4 more than the other, which can be expressed as the equation: x = y + 4
The sum of twice the larger number and five times the smaller number is 120:
![\[ 2x + 5y = 120 \]](https://img.qammunity.org/2022/formulas/mathematics/high-school/hc1r8ja0j6gnno83elximfm508ub7uwryx.png)
Now, substitute the expression for x from the first equation into the second equation:
![\[ 2(y + 4) + 5y = 120 \]](https://img.qammunity.org/2022/formulas/mathematics/high-school/2i6n3e9exb7s4z40qwswwxvgdsy49xzldx.png)
Distribute and combine like terms:
![\[ 2y + 8 + 5y = 120 \]](https://img.qammunity.org/2022/formulas/mathematics/high-school/phcruk8kj1kc6uz0d0gobcesua2vmszhqk.png)
![\[ 7y + 8 = 120 \]](https://img.qammunity.org/2022/formulas/mathematics/high-school/nj5qq4w2tjc2wgg6jtyzd03xz53yzv17fz.png)
Subtract 8 from both sides:
![\[ 7y = 112 \]](https://img.qammunity.org/2022/formulas/mathematics/high-school/jjkg2tdnskxn127zu2dwbv1rrijb0ful97.png)
Divide by 7:
y=16
Now, substitute the value of y back into the equation for x:
![\[ x = y + 4 = 16 + 4 = 20 \]](https://img.qammunity.org/2022/formulas/mathematics/high-school/b55h3br4gi25azpn4fr7dk0zue74yz398e.png)
So, the correct solution is
and
The larger number is 20.