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Chapter 3: Problem 8

Write the ratio in simplest form. 77 to 55

Short Answer

Expert verified
The simplest form of the ratio 77 to 55 is 7 to 5

Step by step solution

01

Determine the GCD

The GCD of 77 and 55 can be determined by listing the factors of each and finding the greatest number that appears in both lists. Factors of 77 are 1, 7, 11, 77; and factors of 55 are 1, 5, 11, 55. Therefore, the GCD of 77 and 55 is 11.
02

Simplify the Ratio

Now, the ratio can be simplified by dividing both numbers by their GCD. Thus, \( \frac{77}{11} = 7 \) and \( \frac{55}{11} = 5 \). Therefore, the simplified form of the ratio is 7 to 5.
03

State the Final Answer

The ratio 77 to 55 in its simplest form is 7 to 5

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

GCD (Greatest Common Divisor)
Understanding the concept of the Greatest Common Divisor (GCD) is crucial for simplifying ratios and fractions effectively. The GCD is the largest number that can divide two or more numbers without leaving a remainder. To find the GCD:
  • First, list all the factors of each number.
  • Identify the common factors shared by the numbers.
  • Choose the largest common factor among them.
For example, in our case of simplifying the ratio 77 to 55, the numbers 77 and 55 both share the factor 11 as their largest common factor. Thus, their GCD is 11.
This step is essential because it sets the basis for simplifying fractions and ratios by reducing them to their simplest form.
Simplifying Fractions
Simplifying fractions is a method used to make fractions easier to work with. It involves reducing a fraction to its smallest equivalent form by dividing the numerator and the denominator by their Greatest Common Divisor (GCD).
Let鈥檚 apply this to simplify the ratio 77 to 55. With the GCD already identified as 11, dividing both numbers by 11 simplifies the fraction, converting 77/55 to 7/5:
  • Divide 77 by 11 to get 7.
  • Divide 55 by 11 to get 5.
Thus, the simplified form of the ratio is 7 to 5. Simplifying fractions or ratios helps in easier computation and clearer understanding, especially when comparing values.
Factoring Numbers
Factoring is breaking down a number into a product of its smaller components or factors. It is a fundamental skill when working with ratios, fractions, or algebraic expressions, as it helps identify the GCD and simplify numbers.
To factor a number:
  • Identify all numbers that can divide the original number without leaving a remainder.
  • These divisors are called factors.
For example, factoring 77, the factors are 1, 7, 11, and 77, since these numbers can all divide 77 without leaving a remainder. Similarly, for 55, the factors are 1, 5, 11, and 55. Factoring is critical as it reveals the GCD, hence enabling the simplification of ratios like in our example.

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Most popular questions from this chapter

You want to paint a piece of pottery. The total price is the cost of the piece plus an hourly painting rate. Studio A sells a vase for \(\$12\) and lets you paint for \(\$7\) an hour. Studio B sells a similar vase for \(\$15\) and lets you paint for \(\$4\) an hour. Which equation would you use to compare the total price at each studio? A. \(7 x-12=4 x-15\) B. \(12+7 x=15+4 x\)

In Exercises 54鈥�56, match the real-life problem with an equation. Then solve the problem. A. \(x+15=7\) B. \(15-x=7\) C. \(15+7=x\) D. \(x+15=-7\) The temperature rose 15 degrees to \(7^{\circ} \mathrm{F}\). What was the original temperature?

GREATEST COMMON FACTOR Find the greatest common factor of the pair of numbers. (Skills Review p. 761 ) $$ 27,36 $$

Solve the equation. Round the result to the nearest hundredth. Check the rounded solution. $$-7 a-9=6$$

Find the unit rate. Earn 126 dollars for working 18 hours
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