Simplify: 2/3 + 4/9 x 7 ½ + 9948/49 x 245.

Simplify: 2/3 + 4/9 x 7 ½ + 9948/49 x 245.

The Correct Answer and Explanation is:

To simplify the expression:23+49×712+994849×245\frac{2}{3} + \frac{4}{9} \times 7\frac{1}{2} + \frac{9948}{49} \times 24532​+94​×721​+499948​×245

Step-by-step Simplification

1. Convert all mixed numbers and perform multiplications first.

Convert 7127\frac{1}{2}721​ to an improper fraction:712=1527\frac{1}{2} = \frac{15}{2}721​=215​

Multiply 49×152\frac{4}{9} \times \frac{15}{2}94​×215​:4×159×2=6018=103\frac{4 \times 15}{9 \times 2} = \frac{60}{18} = \frac{10}{3}9×24×15​=1860​=310​

Multiply 994849×245\frac{9948}{49} \times 245499948​×245:

Note that 245=5×49245 = 5 \times 49245=5×49. So:994849×245=994849×5×49=9948×5=49740\frac{9948}{49} \times 245 = \frac{9948}{49} \times 5 \times 49 = 9948 \times 5 = 49740499948​×245=499948​×5×49=9948×5=49740

Now add everything:23+103+49740=123+49740=4+49740=49744\frac{2}{3} + \frac{10}{3} + 49740 = \frac{12}{3} + 49740 = 4 + 49740 = \boxed{49744}32​+310​+49740=312​+49740=4+49740=49744​

Explanation

The original expression involves both fractions and whole numbers. To simplify accurately, it is essential to follow the order of operations, often remembered as PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

We begin by identifying all operations involved. The expression includes multiplication and addition. According to the rules of arithmetic, multiplication takes precedence over addition, so we perform those calculations first.

First, convert the mixed number 7127\frac{1}{2}721​ into an improper fraction. This is done by multiplying the whole number by the denominator and then adding the numerator. 7×2+1=157 \times 2 + 1 = 157×2+1=15, so 712=1527\frac{1}{2} = \frac{15}{2}721​=215​. Now we multiply 49\frac{4}{9}94​ by 152\frac{15}{2}215​. Multiplying the numerators and denominators gives us 6018\frac{60}{18}1860​, which simplifies to 103\frac{10}{3}310​.

Next, we handle 994849×245\frac{9948}{49} \times 245499948​×245. Here, recognizing that 245 is divisible by 49 makes the calculation simpler. Since 245=5×49245 = 5 \times 49245=5×49, the multiplication simplifies: 994849×5×49\frac{9948}{49} \times 5 \times 49499948​×5×49. The 49s cancel out, leaving 9948×5=497409948 \times 5 = 497409948×5=49740.

Now add all the parts: 23+103=123=4\frac{2}{3} + \frac{10}{3} = \frac{12}{3} = 432​+310​=312​=4. Adding this to 49740 gives the final result:4+49740=497444 + 49740 = \boxed{49744}4+49740=49744​

This process emphasizes the importance of simplifying step by step, recognizing number patterns, and respecting the proper order of operations.