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Zorluk: Çok zorHCF and LCM

Three storage vessels in a chemical laboratory contain 1235\frac{12}{35} litres, 1825\frac{18}{25} litres, and 2449\frac{24}{49} litres of different reagents. What is the maximum capacity (in litres) of a single measuring container that can be used to measure the exact quantity of reagent from any of the three vessels without leaving any remainder?

  1. 61225\frac{6}{1225} litresCevap
  2. B
    721225\frac{72}{1225} litres
  3. C
    12256\frac{1225}{6} litres
  4. D
    7272 litres

Cevap

The maximum capacity of the measuring container is 61225\frac{6}{1225} litres.
To find the largest measuring unit that can measure fractional quantities exactly, we need the HCF of those fractions. The HCF of fractions is defined as HCF of numeratorsLCM of denominators\frac{\text{HCF of numerators}}{\text{LCM of denominators}}. Calculating HCF(12,18,24)=6\text{HCF}(12, 18, 24) = 6 and LCM(35,25,49)=1225\text{LCM}(35, 25, 49) = 1225 yields 61225\frac{6}{1225} litres.

Adım Adım Çözüm

1
Identify the mathematical rule required for measuring fractional quantities exactly.
To find the maximum capacity of a container that measures given fractional quantities an exact number of times, compute the HCF of the fractions: HCF(ab,cd,ef)=HCF(a,c,e)LCM(b,d,f)\text{HCF}\left(\frac{a}{b}, \frac{c}{d}, \frac{e}{f}\right) = \frac{\text{HCF}(a, c, e)}{\text{LCM}(b, d, f)}.
The measuring container's capacity must be a common divisor of all three fractional volumes.
2
Calculate the HCF of the numerators 1212, 1818, and 2424.
HCF(12,18,24)=6\text{HCF}(12, 18, 24) = 6.
The prime factorizations are 12=22×312 = 2^2 \times 3, 18=2×3218 = 2 \times 3^2, and 24=23×324 = 2^3 \times 3. The highest common power of prime factors shared by all three numbers is 21×31=62^1 \times 3^1 = 6.
3
Calculate the LCM of the denominators 3535, 2525, and 4949.
LCM(35,25,49)=1225\text{LCM}(35, 25, 49) = 1225.
The prime factorizations are 35=5×735 = 5 \times 7, 25=5225 = 5^2, and 49=7249 = 7^2. The LCM takes the highest power of each prime: 52×72=25×49=12255^2 \times 7^2 = 25 \times 49 = 1225.
4
Substitute the calculated HCF of numerators and LCM of denominators into the fraction HCF formula.
HCF=61225\text{HCF} = \frac{6}{1225} litres.
Dividing the HCF of the numerators by the LCM of the denominators gives the exact maximum volume.

Anahtar Kavram

HCF of Fractions
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Konular

HCF and LCM

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