What is the value of $\frac{13 \times 4-26}{54-7 \times 4}$ ?

Question

Prepp Team · Accepted Answer

Evaluating the Mathematical Expression

We are asked to find the value of the given mathematical expression, which is a fraction:

$$ \frac{13 \times 4-26}{54-7 \times 4} $$

To evaluate this expression, we need to follow the order of operations (BODMAS or PEMDAS), which dictates the sequence of calculations: Brackets/Parentheses, Orders/Exponents, Division and Multiplication (from left to right), Addition and Subtraction (from left to right).

Step 1: Evaluate the Numerator

The numerator is $13 \times 4 - 26$. According to the order of operations, we perform multiplication before subtraction.

First, calculate the product: $13 \times 4$
$13 \times 4 = 52$
Next, perform the subtraction: $52 - 26$
$52 - 26 = 26$

So, the value of the numerator is 26.

Step 2: Evaluate the Denominator

The denominator is $54 - 7 \times 4$. Again, we perform multiplication before subtraction.

First, calculate the product: $7 \times 4$
$7 \times 4 = 28$
Next, perform the subtraction: $54 - 28$
$54 - 28 = 26$

So, the value of the denominator is 26.

Step 3: Evaluate the Fraction

Now we have the value of the numerator and the denominator. The fraction becomes:

$$ \frac{\text{Numerator}}{\text{Denominator}} = \frac{26}{26} $$

Finally, we perform the division:

$$ \frac{26}{26} = 1 $$

Therefore, the value of the expression $\frac{13 \times 4-26}{54-7 \times 4}$ is 1.

The steps can be summarized as follows:

$$ \frac{(13 \times 4) - 26}{54 - (7 \times 4)} $$

$$ = \frac{52 - 26}{54 - 28} $$

$$ = \frac{26}{26} $$

$$ = 1 $$

Revision Table: Order of Operations (BODMAS/PEMDAS)

Letter	Meaning	Operation Type	Order
B / P	Brackets / Parentheses	Calculations inside grouping symbols	Perform in this order from top to bottom. Division and Multiplication are done from left to right. Addition and Subtraction are done from left to right.
O / E	Orders / Exponents	Powers and Roots
D / M	Division / Multiplication	Division and Multiplication
A / S	Addition / Subtraction	Addition and Subtraction

Additional Information: Simplifying Fractions

After evaluating the numerator and denominator, we get a fraction $\frac{26}{26}$. A fraction represents division. When the numerator and the denominator of a fraction are the same non-zero number, the value of the fraction is 1.

For example:

$\frac{5}{5} = 1$
$\frac{100}{100} = 1$
$\frac{-7}{-7} = 1$

Simplifying a fraction involves dividing both the numerator and the denominator by their greatest common divisor (GCD). In the case of $\frac{26}{26}$, the GCD of 26 and 26 is 26. Dividing both by 26 gives $\frac{26 \div 26}{26 \div 26} = \frac{1}{1} = 1$.