2416 /www/nrich/html/content/id/2416/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p><mdo:image alt="chocolate" src="icon.png" style="float: left; width: 100px; height: 100px;"></mdo:image>George and Jim want to buy a chocolate bar.</p> <p>George needs 2p more and Jim needs 50p more to buy it.</p> <p>When they put their money together, it is still not enough to pay for the chocolate bar.</p> <p> </p> <p><strong>How much is the chocolate bar?</strong></p></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <span class="editorial">You answered this tricky question very well using several different methods. Jack from Beech Primary wrote</span> :<br></br> <br></br> <p>Because Jim needs 50p more to buy the chocolate, we know the chocolate costs at least 50p.</p> <p>I used trial and improvement to find the answer.</p> We said C is the chocolate bar, J is Jim and G is George. <div style="clear: both;"> <p>If the chocolate (C) costs 100p, then Jim has 50p and George has 98p. But the total would be 148p and this is too much, because they say that Jim and George do not have enough money when it's put together.</p> <div style="clear: both;"> <p>This means the chocolate bar is less than 100p but more than 50p.</p> <div style="clear: both;"></div> <table cellspacing="1" border="1"> <tbody> <tr> <td>C</td> <td>J</td> <td>G</td> <td>Total</td> <td></td> </tr> <tr> <td>100p</td> <td>50p</td> <td>98p</td> <td>148p</td> <td>Too much</td> </tr> <tr> <td>70p</td> <td>20p</td> <td>68p</td> <td>88p</td> <td>Too much</td> </tr> <tr> <td>50p</td> <td>0p</td> <td>48p</td> <td>48p</td> <td>Could be right</td> </tr> <tr> <td>51p</td> <td>1p</td> <td>49p</td> <td>50p</td> <td>Could be right</td> </tr> <tr> <td>52p</td> <td>2p</td> <td>50p</td> <td>52p</td> <td>Too much</td> </tr> <tr> <td>53p</td> <td>3p</td> <td>51p</td> <td>54p</td> <td>Too much</td> </tr> </tbody> </table> <div style="clear: both;"></div> <p>So the right answer is either 50p or 51p. If Jim definitely has some money, then the cost of the chocolate bar is 51p.</p> <p class="editorial">Well done, Jack. Luke from St. Nicolas Junior School, Newbury reasoned his way through the solution :</p> <p style="clear: both;">George needs 2p more, so Jim must have less than 2p because otherwise, he would have given George the 2p he needed. So I chose 1p for Jim. As Jim needed 50p more, the chocolate bar must cost 51p.</p> <div style="clear: both;"></div> <table cellspacing="1" border="1"> <tbody> <tr> <td>Jim</td> <td>Cost of chocolate bar</td> <td>George</td> </tr> <tr> <td>1p</td> <td>51p</td> <td>49p</td> </tr> </tbody> </table> <div style="clear: both;"></div> <p style="clear: both;">Add George's money and Jim's money, which is 50p, so they still don't have enough to buy the chocolate bar together. Another solution I found was:</p> <div style="clear: both;"></div> <table cellspacing="1" border="1"> <tbody> <tr> <td>Jim</td> <td>Cost of chocolate bar</td> <td>George</td> </tr> <tr> <td>0p</td> <td>50p</td> <td>48p</td> </tr> </tbody> </table> </div> </div> <p>This works but the words of the question make me think that Jim must have some money, that is 1p.</p> <p class="editorial">Very well explained, Luke. Finally, Mikey from Archbishop of York C of E School used algebra to arrive at his solution:</p> <div>Let's say George has G pence then choc bar costs G+2 pence.</div> <div>Say C = G + 2 or G = C - 2. <div style="clear: both;">Similarly Jim has J pence and choc costs J + 50 pence.</div> <div style="clear: both;">Say C = J + 50 or J = C - 50.</div> <div style="clear: both;">If their total money is less than cost of a bar then G + J is less than choc bar.</div> <div style="clear: both;">So G + J &lt; C which is the same as (C - 2) + (C - 50) &lt; C.</div> <div style="clear: both;">This simplifies to 2C - 52 &lt; C and further to C &lt; 52.</div> <div style="clear: both;">So the choc bar costs less than 52p but George has 48p more than Jim.</div> <div style="clear: both;">So the only possibility is for George to have 49p, Jim 1p, their total is 50p and the choc bar must be 51p (as we showed above it must be less than 52p).</div> <p class="editorial" style="clear: both;">Another clear solution - fantastic!</p> </div> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">This problem supports children in calculating with money but also requires them to be systematic. Playing with the problem for a few minutes thinking about what the minimum price of the chocolate bar could be is a useful starting point. How can pupils convince each other that they have all the solutions? Although simultaneous equations seem to be appropriate they would just get in the way!<br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p>Try with a few different amounts of money for the two children. Does this help you to solve the problem?</p> <p>What is the least the chocolate bar could be?</p> <p>How much does this mean each child would have?</p> <p>Are there any more answers and how do you know?</p> <br></br></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> The chocolate bar is 51 p (they have 49 p and 1 p respectively). <br></br><br></br></mdoxml> 2 4 0 1 0 0 0 George and Jim want to buy a chocolate bar. George needs 2p more and Jim need 50p more to buy it. How much is the chocolate bar? Decision Mathematics and Combinatorics Combinations Using, Applying and Reasoning about Mathematics Working systematically Measures and Mensuration Money Calculations and Numerical Methods Mental addition & subtraction