To simplify the ratio (2 - \frac{26}{27} : \frac{39}{45}), let's break it down step by step.
First, simplify (2 - \frac{26}{27}):
Convert (2) to a fraction with a denominator of (27):
[
2 = \frac{54}{27}
]
Now subtract:
[
\frac{54}{27} - \frac{26}{27} = \frac{54 - 26}{27} = \frac{28}{27}
]
Next, simplify (\frac{39}{45}):
- Find the greatest common divisor (GCD) of (39) and (45). The GCD is (3).
- Divide both the numerator and the denominator by (3):
[
\frac{39 \div 3}{45 \div 3} = \frac{13}{15}
]
Now we can rewrite the ratio:
[
\frac{28}{27} : \frac{13}{15}
]
To express this ratio in a more usual form, we cross-multiply:
[
28 \times 15 : 27 \times 13
]
Calculating each term:
- (28 \times 15 = 420)
- (27 \times 13 = 351)
Thus, the ratio simplifies to:
[
420 : 351
]
To further simplify, find the GCD of (420) and (351). The GCD is (3).
Now divide both sides by (3):
[
\frac{420 \div 3}{351 \div 3} = 140 : 117
]
So, the final simplified ratio is:
[
\mathbf{140 : 117}
]
