Grandfather is #15# time the age of his grandson today, in #10# years the grandson will be #3/17# the age of the grandfather. How old are the grandfather and grandson?

Question

Aurora Arias · Accepted Answer

Grandfather is #65# years old.
Grandson is #5# years old.

Let the grandfather's age be #x# and the grandson's age be #y#. We are told #color(white)("XXX")x=15y# and #color(white)("XXX")y+10=3/17(x+10)# #color(white)("XXXXXX")rarr 17y+170=3x+30# #color(white)("XXXXXXXXXXXXXX")=45y+30# (since #x=15y#) #color(white)("XXXXXX")rarr140=28y# #color(white)("XXXXXX")rarry=5# #color(white)("XXX")x=15y=65#

Roman Alvarez · Answer

undefined Let's denote the current age of the grandson as $ G $ and the current age of the grandfather as $ G \times 15 $. In 10 years, the age of the grandson will be $ G + 10 $, and the age of the grandfather will be $ G \times 15 + 10 $.

According to the second statement, in 10 years, the age of the grandson will be $ \frac{3}{17} $ of the age of the grandfather:

$$ G + 10 = \frac{3}{17} \times (G \times 15 + 10) $$

Expanding this equation:

$$ 17(G + 10) = 3(G \times 15 + 10) $$

$$ 17G + 170 = 45G + 30 $$

$$ 170 - 30 = 45G - 17G $$

$$ 140 = 28G $$

$$ G = 5 $$

So, the grandson is currently 5 years old. The grandfather is $ G \times 15 = 5 \times 15 = 75 $ years old.