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25 Naomi travels to an interview.
She travels by car for
(1)/(3) of the journey.
She travels by train for
(5)/(8) of the journey.
She walks for the remaining
500m of the journey.
Find the length of this journey in kilometres.

$25$ Naomi travels to an interview. $\newline$ She travels by car for $\frac{1}{3}$ of the journey. $\newline$ She travels by train for $\frac{5}{8}$ of the journey. $\newline$ She walks for the remaining $500 \mathrm{~m}$ of the journey. $\newline$ Find the length of this journey in kilometres.

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Add and subtract rational expressions

Full solution

Q.

25

Naomi travels to an interview.

\newline

She travels by car for

\frac{1}{3}

of the journey.

\newline

She travels by train for

\frac{5}{8}

of the journey.

\newline

She walks for the remaining

500 \mathrm{~m}

of the journey.

\newline

Find the length of this journey in kilometres.

Add Fractions: First, let's add up the fractions of the journey that Naomi travels by car and train. $\newline$ $(1)/(3) + (5)/(8)$ $\newline$ To add these fractions, we need a common denominator.
Convert to Common Denominator: The least common denominator (LCD) for $3$ and $8$ is $24$ . So, we convert the fractions. $(1)/(3) = (8)/(24)$ and $(5)/(8) = (15)/(24)$ Now, we add them up. $(8)/(24) + (15)/(24) = (23)/(24)$
Calculate Walking Distance: Now we know that Naomi travels $(\frac{23}{24})$ of the journey by car and train. $\newline$ The remaining part of the journey is the $500$ m she walks.
Write Total Journey Equation: To find the total journey, we need to express the walking distance as a fraction of the total journey. $\newline$ Since we don't know the total journey yet, let's call it ' $x$ ' kilometers. $\newline$ $500\,\text{m}$ is $0.5$ kilometers (because $1000\,\text{m} = 1\,\text{km}$ ). $\newline$ So, the walking part is $\frac{0.5}{x}$ of the journey.
Isolate 'x' in Equation: Now we can write an equation for the total journey. $\newline$ $(\frac{23}{24}) + \frac{0.5}{x} = 1$ (because the car, train, and walking parts add up to the whole journey)
Multiply by 'x': To solve for 'x', we need to isolate it on one side of the equation. $\newline$ $\frac{0.5}{x} = 1 - \frac{23}{24}$ $\newline$ $\frac{0.5}{x} = \frac{24}{24} - \frac{23}{24}$ $\newline$ $\frac{0.5}{x} = \frac{1}{24}$
Find Total Distance: Now we multiply both sides by ' $x$ ' to get rid of the fraction. $\newline$ $0.5 = \frac{1}{24} \times x$
Find Total Distance: Now we multiply both sides by $x$ to get rid of the fraction. $0.5 = \frac{1}{24} \times x$ To find $x$ , we divide both sides by $\frac{1}{24}$ . $x = \frac{0.5}{\frac{1}{24}}$ $x = 0.5 \times \frac{24}{1}$ $x = 12$

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