Rizwan writes down three numbers a,b and C a: b=1:3 b:c=6:5 (a) (i) Find a:b:c (ii) Express a as a fraction of the total of the three numbers a, b and C Emma writes down three numbers m, n and p n=2m p=5n (b) Find m:p

Let's begin with your questions.<br /><br />(a) <br />Given that:<br />i) $a: b = 1: 3 $ <br />ii) $ b: c = 6: 5 $<br /><br />We need to find:<br /><br />(i) The ratio ${a:b:c}$<br />(ii) The ratio of ${a}$ to the total of the numbers a, b and c.<br /><br />Sol:<br /><br />(i)<br />First, we're given ${ a: b = 1: 3} $. This states that for every one "part" of "a", we have three "parts" of "b". <br />Similarly, we're given ${ b: c = 6: 5} $, stating that for every six "parts" of "b", we have five "parts" of "c".<br /><br />To find the proportion between a, b and c, we can set the value of "b" equal in both ratios. We can uniformly multiply a: b by 6 and b: c by 3, which will maintain our ratios while setting b to be consistent across both:<br /><br />so we get: <br />${a: b = 1*6: 3*6 = 6: 18}$ <br />and <br />${b: c = 6*3: 5*3 = 18: 15}$ <br /><br />So, the final ${a:b:c=6:18:15}.$ <br /><br />(ii) <br />Now we express "$a$" as a fraction of the total of the three numbers "${a, b, c}$<br />Following from our results in part i, we have accounted for 6 "parts" of a, 18 "parts" of b, and 15 "parts" of c. <br /><br />Therefore ${a} ${as} ${a}$ ${fraction}$ ${of}$ ${a+b+c} =<br />$= \frac{a}{a+b+c}<br />= \frac{6}{6 + 18 + 15}= \frac{6}{39}$<br /><br />Next let's move on to the part b of your problem:<br />\[<br />\begin{array}{l}<br />n=2 m\\<br />p=5 n<br />\end{array}<br />\]<br /><br />We need to find ${m:p}$. <br />Given equations are <br />"n" = 2 "m" <br />and "p" = 5 "-n"<br /><br />By substituting "n" into the second equation, we get:<br /> "p" = 5 (${2m}$) which gives "p" = ${10m}$<br /><br />So the ratio of ${m}$ to ${p = m: p = 1:10}$ <br /><br />So, here is the final results from our calculations: <br />(a) <br />(i)-${a:b:c = 6:18:15}$<br />(ii)-${a/(a+b+c) = 6/39}$<br />(b) ${(m:p=1:10)}$ <br /><br />I hope these derivations were clear and met the goals of your learning guide! Feel free to keep the questions coming to fill the holes in your knowledge!