# Daughter 2849

Dad, mom, and daughter are 100 years old. A father is four times as old as a daughter. Mom is ten years older than half the father and daughter. How old are they?

d =  12
m =  40
o =  48

### Step-by-step explanation:

o+m+d=100
o = 4d
m=10+(o+d)/2

o+m+d=100
o = 4·d
m=10+(o+d)/2

d+m+o = 100
4d-o = 0
d-2m+o = -20

Pivot: Row 1 ↔ Row 2
4d-o = 0
d+m+o = 100
d-2m+o = -20

Row 2 - 1/4 · Row 1 → Row 2
4d-o = 0
m+1.25o = 100
d-2m+o = -20

Row 3 - 1/4 · Row 1 → Row 3
4d-o = 0
m+1.25o = 100
-2m+1.25o = -20

Pivot: Row 2 ↔ Row 3
4d-o = 0
-2m+1.25o = -20
m+1.25o = 100

Row 3 - 1/-2 · Row 2 → Row 3
4d-o = 0
-2m+1.25o = -20
1.875o = 90

o = 90/1.875 = 48
m = -20-1.25o/-2 = -20-1.25 · 48/-2 = 40
d = 0+o/4 = 0+48/4 = 12

d = 12
m = 40
o = 48

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