Question

sted in the chart below.StudentJamesGilbertMatthewSimonTime2WIN5(minutes)612Find the difference between the fastest boy's time and the slowestboy'stimeDriffrence is 11/12 - 7/12 =4/12fastest =7/12 ( Simon)slowest =11/12 ( Gilbert)2. To break the school's record, the boys' time had to be faster than12minutes. Estimate to decide if the boys broke the record. If so, howmuch faster were they? If not, how much slower were they?

Answer 1

We have the time in minutes for each of the students in the question:

• James ---> 2/3

,

• Gilbert ---> 11/12

,

• Matthew ---> 5/6

,

• Simon ---> 7/12

We can order these times in ascending order as follows:

$\frac{7}{12}<\frac{2}{3}<\frac{5}{6}<\frac{11}{12}$

Therefore, Simon was the fastest in the race, then James, Matthew, and Gilbert.

We can use decimals to see it easier as follows:

$\begin{gathered} \frac{7}{12}\approx0.583333333333 \\ \frac{2}{3}\approx0.666666666667 \\ \frac{5}{6}\approx0.833333333333 \\ \frac{11}{12}\approx0.916666666667 \end{gathered}$

The school record is:

$\begin{gathered} 2\frac{7}{12}=2+\frac{7}{12}=\frac{(2)(12)+(1)(7)}{12}=\frac{24+7}{12}=\frac{31}{12} \\ \frac{31}{12}\approx2.58333333333 \end{gathered}$

As we can see all of the students broke the school's record. All of the times are less than the one of the school's record for that race.

To find how much faster were they, we can subtract each of the fractions as follows:

For Simon:

$\frac{31}{12}-\frac{7}{12}=\frac{24}{12}=2$

Simon was 2 minutes less than the School record.

For James:

$\frac{31}{12}-\frac{2}{3}=\frac{31}{12}-\frac{4\cdot2}{4\cdot3}=\frac{31}{12}-\frac{8}{12}=\frac{23}{12}$

James was 23/12 minutes less than the School's record.

For Matthew

$\frac{31}{12}-\frac{5}{6}=\frac{31}{12}-\frac{2\cdot5}{2\cdot6}=\frac{31}{12}-\frac{10}{12}=\frac{21}{12}=\frac{\frac{21}{3}}{\frac{12}{3}}=\frac{7}{4}$

Matthew was 21/12 minutes less than the School's record or, equivalently, 7/4 minutes less than the School's record.

For Gilbert

$\frac{31}{12}-\frac{11}{12}=\frac{20}{12}=\frac{\frac{20}{4}}{\frac{12}{4}}=\frac{5}{3}$

Gilbert was 20/12 minutes less than the School's record. We can also say that Gilbert was 5/3 minutes less than the School's record. In fact, 20/12 and 5/3 are equivalent fractions.

In summary, we can say:

First, all of the students broke the School's record, and they were faster in the following way:

• Simon was 2 minutes less than the School record.

,

• James was 23/12 minutes less than the School's record.

,

• Matthew was 21/12 (7/4) minutes less than the School's record or, equivalently, 7/4 minutes less than the School's record.

,

• Gilbert was 20/12 (5/3) minutes less than the School's record.,