Use the properties of operations and rules for multiplying and dividing to solve:
( -3) divided by (3/4) times (2/7) times (-5/16 divided by (5/7)
To solve this, we will simplify each of the fractions and then carry out the multiplication and division operations.
First, simplify the fractions:
(-3) ÷ (3/4) = (-3) × (4/3) = -12/3 = -4
(-5/16) ÷ (5/7) = (-5/16) × (7/5) = -35/80 = -7/16
Now, multiply the simplified fractions:
(-4) × (2/7) × (-7/16) = -8/112 = -1/14
So, (-3) divided by (3/4) times (2/7) times (-5/16) divided by (5/7) equals -1/14.
To solve this expression, follow the order of operations: parentheses, multiplication/division (from left to right), and addition/subtraction (from left to right).
Step 1: Simplify the expression within parentheses:
The expression (-3) divided by (3/4) can be simplified by multiplying -3 by the reciprocal (or inverse) of 3/4. The reciprocal of 3/4 is 4/3, so we have:
(-3) divided by (3/4) = -3 × (4/3)
Step 2: Simplify the expression involving division by multiplying the reciprocals:
The expression (-5/16) divided by (5/7) can be simplified by multiplying -5/16 by the reciprocal of 5/7. The reciprocal of 5/7 is 7/5, so we have:
(-5/16) divided by (5/7) = -5/16 × (7/5)
Now, let's substitute the simplified expressions back into the original expression:
(-3) divided by (3/4) times (2/7) times (-5/16 divided by (5/7) becomes:
-3 × (4/3) × (2/7) × (-5/16) × (7/5)
Step 3: Multiply in any order:
Using the properties of multiplication, we can multiply the numerator (top) of one fraction by the numerator of another fraction and the denominator (bottom) of one fraction by the denominator of another fraction. We can multiply the numbers straight across.
-3 × (4/3) × (2/7) × (-5/16) × (7/5) =
-3 × 4 × 2 × -5 × 7 / 3 × 7 × 16 × 5
Step 4: Simplify the expression by canceling common factors:
When we multiply opposite signs, the product is negative. Cancel out common factors between the numerator and denominator to reduce the expression:
-(-3) × 4 × 2 × 5 × 7 / 3 × 7 × 16 × 5 =
3 × 4 × 2 × 7 / 3 × 7 × 16
Step 5: Continue simplifying:
Cancel factors where possible:
3 × 4 × 2 × 7 / 3 × 7 × 16 =
6 × 7 / 16 =
42 / 16
Step 6: Simplify or express as a mixed number:
The fraction 42/16 can be simplified by finding the greatest common divisor (GCD) of 42 and 16, which is 2. Divide both the numerator and denominator by 2:
42/16 = (42 ÷ 2) / (16 ÷ 2) = 21/8
So, the expression (-3) divided by (3/4) times (2/7) times (-5/16 divided by (5/7) simplifies to 21/8.
To solve this expression step-by-step, we will use the properties of operations and rules for multiplying and dividing.
Step 1: Simplify the division of fractions within parentheses.
Start with the expression:
(-5/16) divided by (5/7)
To divide fractions, we invert (or flip) the second fraction and multiply.
=(-5/16) multiplied by (7/5)
Step 2: Multiply fractions within parentheses.
Multiply the other fractions within parentheses:
=(-3) divided by (3/4) multiplied by (2/7) multiplied by (-5/16) multiplied by (7/5)
Step 3: Simplify multiplication and division from left to right.
Multiply and divide from left to right:
=(-3/1) multiplied by (2/1) multiplied by (-5/1) multiplied by (7/1) multiplied by (-5/16) multiplied by (7/5)
Step 4: Simplify the expression by canceling out common factors.
Let's simplify the numerators and denominators separately to cancel out any common factors:
=(-3 x 2 x -5 x 7)/ (1 x 1 x 1 x 1 x 16 x 5)
Step 5: Simplify the numerator and denominator further.
Calculate the product of the numerators and denominators:
=210/80
Step 6: Reduce the fraction to its simplest form.
We can simplify the fraction by dividing both the numerator and denominator by their greatest common divisor, which is 10:
=21/8
Therefore, the expression (-3) divided by (3/4) times (2/7) times (-5/16) divided by (5/7) simplifies to 21/8.