Explain in your own words how to find the rational number th

Explain in your own words how to find the rational number that is one-fourth of the way between two different rational numbers. Then give two rational numbers and ask your classmates to find a rational number that is between the two given rational numbers. Explain how to simplify a squareroot . Describe what it means to rationalize the denominator. Post one expression for your classmates to either simplify a squareroot or rationalize the denominator. Explain in your own words how to multiply decimals. Explain in your own words how to divide decimals. Post a decimal expression for you classmates to either add or subtract. Explain how to convert a decimal to a percent, a percent to a decimal, and a fraction to a percent. Post a percent, fraction, or decimal for your classmates to convert.

Solution

Let p/q and r/s be two rational numbers in simplest form ( the numerator and denominator have no common factors). Also let p/q < r/s. The difference between these two rational numbers is ( r/s - p/q) = ( qr -ps)/qs. The rational number that is one-fourth of the way between these two rational numbers is p/q + 1/4(qr-ps)/qs = [4ps+(qr -ps)]/4qs = (3ps +qr)/4qs.

If the above example is too abstract, let us consider the numbers 1/2 and 2/3. We know that 1/2 ( = 2/4) < 2/3. Also, 2/3 - 1/2 = (4 -3)/6 = 1/6. The rational number that is one-fourth of the way between 1/2 and 2/3 is 1/2 + 1/4(1/6) = 1/2 + 1/24 = (12+1)/24 = 13/24.

Given two rational numbers p/q and r/s in their simplest form, the rational number (3ps +qr)/4qs. is between these two rational numbers. Let us find another rational number between these two rational numbers. We know that r/s - p/q = ( qr -ps)/qs . The number p/q + 1/2( qr -ps)/qs = p/q + (qr -ps)/2qs = [2ps + (qr -ps)]/2qs = (ps+qr)/2qs is also between the rational numbers p/q and r/s. Further, if we again take the example of the rational numbers 1/2 and 2/3, then we know from the abobe that 2/3 - 1/2 = 1/6. Then 1/2 + 1/2(1/6) = 1/2 + 1/12 = (6 +1)/12 = 7/12 is another rational number between 1/2 and 2/3. To generalise, if we add a fraction (less than 1) of the difference of two rational numbers to the smaller of the two rational number, we will get a rational number between the two given rational numbers.