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Junior Mathematics Competition 2018 Solutions and Comments

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Junior Mathematics Competition 2018 Solutions and Comments Department of Mathematics and Statistics Junior Mathematics Competition 2018 Solutions and Comments Web: maths.otago.ac.nz/jmc Twitter: @uojmc YEAR 9 (FORM 3) PRIZE WINNERS FIRST GrACE WU ST CUTHBERT’S COLLEGE SECOND JAMES (He) XU KRISTIN SCHOOL THIRD MICHAEL Li MACLEANS COLLEGE TOP 30 (IN SCHOOL ORDER): DaVID EVANs, CANTERBURY HOME EDUCATORS HAMISH KERR, HAMILTON BOYS’ HIGH SCHOOL SAMUEL DaRR, HUTT INTERMEDIATE SCHOOL RHIANNON MACKIe, HUTT VALLEY HIGH SCHOOL MARTIN BROOK, JOHN MCGlASHAN COLLEGE RANITHU SENDIV RODRIGo, JOHN PaUL COLLEGE WESLEY NGo, KINGS COLLEGE ANDREW YU, KINGS COLLEGE CATHERINE ZHANG, LyNଏELD COLLEGE KEVIN LiN, MACLEANS COLLEGE WEI TIAN Teo, MACLEANS COLLEGE HOSEA TONG-Ho, MACLEANS COLLEGE REGINA YUN, MACLEANS COLLEGE NANCY CHEN, MACLEANS COLLEGE JESSICA HONG, MACLEANS COLLEGE JAMES HUI, MACLEANS COLLEGE KERWIN MAAss, NEWLANDS COLLEGE ZIYUE (CLAIRe) XU, NORTHCROSS INTERMEDIATE HENRY ZASLOW, ONSLOW COLLEGE LuCY AITKEN, RANGI RURU GiRLS’ SCHOOL JESSICA WANG, RANGITOTO COLLEGE GrACE PuI, ST CUTHBERT’S COLLEGE EMMA YiNG, ST CUTHBERT’S COLLEGE BRENA MERZ, ST CUTHBERT’S COLLEGE BRADEN Dye, ST KENTIGERN COLLEGE MAX FENG, ST KENTIGERN COLLEGE TONY SHI, WESTLAKE BOYs’ HIGH SCHOOL YEAR 10 (FORM 4) PRIZE WINNERS FIRST MICHAEL KENNEDY LiSTON COLLEGE SECOND OLIVER DaI MACLEANS COLLEGE THIRD JAMES SUTTON RONGOTAI COLLEGE TOP 30 (IN SCHOOL ORDER): ALDOUS SCHOLLUM, AVONDALE COLLEGE SHREYA SRIRAM, AVONDALE COLLEGE DuNCAN HAMILTON, CHRISTCHURCH BOYS’ HIGH SCHOOL HARRY NEIL, HILLCREST HIGH SCHOOL TOBIAS DeVEREUX, KAVANAGH COLLEGE JASON Lee, KINGS COLLEGE ERIC ZHANG, KRISTIN SCHOOL ANNIE PenG, KRISTIN SCHOOL MAGGIE CHEN, MACLEANS COLLEGE RICK HAN, MACLEANS COLLEGE LiSA SANDBROOK, MACLEANS COLLEGE JEREMY HUANG, MACLEANS COLLEGE ANNA Li, OREWA COLLEGE OLIVER LiN, PaKURANGA COLLEGE KIKI SU, PaKURANGA COLLEGE SEOHYUN KIM, RANGITOTO COLLEGE TONY SCHAUFELBERGER, RUTHERFORD COLLEGE JET SHAND-Pease, SELWYN COLLEGE ARISA MORI, ST ANDREW’S COLLEGE JENNY YUAN, ST CUTHBERT’S COLLEGE JUSTIN XIANG, ST KENTIGERN COLLEGE NATHAN XU, ST KENTIGERN COLLEGE MICHELLE GuAN, ST KENTIGERN COLLEGE EVAN METCALFe, ST KENTIGERN COLLEGE HAOKUN He, ST KENTIGERN COLLEGE PhILLIP HAN, WESTLAKE BOYs’ HIGH SCHOOL TAEWON YUN, WESTLAKE BOYS’ HIGH SCHOOL YEAR 11 (FORM 5) PRIZE WINNERS FIRST EQUAL LuKE YING FENG BAO AUCKLAND GrAMMAR SCHOOL FIRST EQUAL CHEN HUAN LiU EPSOM GiRLS’ GrAMMAR SCHOOL THIRD TERRY SHEN MACLEANS COLLEGE TOP 30 (IN SCHOOL ORDER): ANDREW JIA AN CHe, AUCKLAND GrAMMAR SCHOOL NATHAN CHEN, AUCKLAND GrAMMAR SCHOOL MACKINLEY HE, AUCKLAND GrAMMAR SCHOOL ANDREW YU AN CHEN, AUCKLAND INTERNATIONAL COLLEGE ISAAC HAo-EN YUAN, AUCKLAND INTERNATIONAL COLLEGE MAX YOUNG, CHRISTCHURCH BOYs’ HIGH SCHOOL VICTORIA SUN, EPSOM GiRLS’ GrAMMAR SCHOOL DaNIEL CLARK, GlENDOWIE COLLEGE EMILY ZOU, GlENDOWIE COLLEGE ISHAN NATH, JOHN PaUL COLLEGE NATHANIEL MASFEN-YAN, KINGS COLLEGE HANBO XIe, KINGS COLLEGE RANUDI LelWALA, MACLEANS COLLEGE JAMIE WU, MACLEANS COLLEGE YUNGE YU, MACLEANS COLLEGE ANGELA YANG, MACLEANS COLLEGE DaRSH CHAUDHARI, MACLEANS COLLEGE KELVIN Gu, MACLEANS COLLEGE RAY WU, NEWLANDS COLLEGE JEREMY ISHI, PaKURANGA COLLEGE YUHAN (LiNDA) TANG, PiNEHURST SCHOOL MINJU KIM, RANGITOTO COLLEGE XAVIER YiN, ST KENTIGERN COLLEGE TONY YU, ST KENTIGERN COLLEGE GrACE CHANG, ST KENTIGERN COLLEGE JAMES ALEXANDER, ST Peter’S COLLEGE (EPSOM) STEVEN WANG, ST Peter’S SCHOOL (CAMBRIDGe) 2018 ANSWERS AS ALWAYS ONLY ONE METHOD IS SHOWN. OFTEN SEVERAL METHODS EXISt. WE DON’T GUARANTEE THAT THE METHOD SHOWN IS THE BEST OR FASTESt. QUESTION 1: 10 MARKS (YEARS 9 AND BELOW ONLY) (A) A RECTANGULAR WALL MEASURES 4.5 M BY 3 M. SHOW THAT THE AREA OF THE WALL IS 13.5 M2. EITHER 4.5 × 3 OR 3 × 4.5. AN EASY START BUT SURPRISINGLY A FEW STUDENTS MISSED IT OUT! (B) SUPPOSE THE AREA OF A ROLL OF WALLPAPER HAS DIMENSIONS 5 M BY 0.5 M. (I) WHAT IS THE AREA COVERED BY ONE ROLL OF WALLPAPER? 5 × 0.5 = 2.5 M2. WELL DONE. (MANY STUDENTS WROTE 2.5 M INSTEAD OF 2.5 M2.) (II) HOW MANY ‘WHOLE’ ROLLS OF WALLPAPER WOULD YOU NEED TO COVER THE WALL IN PART (A)? YOUR ANSWER SHOULD NOT HAVE A FRACTION OR DECIMAL COMPONENT IN It. 13.5 / 2.5 = 5.4 SO 6 ROLLS ARE NEEDED. MANY STUDENTS ROUNDED ‘DOWN’ TO 5, LEAVING PART OF THE WALL BARE OF WALLPAPER. REMEMBER THAT IS GENERALLY BETTER TO BUY A LITTLE TOO MUCH THAN NOT ENOUGH! (C) IF A WINDOW COVERS 2.7 M2 ON THE WALL, WHAT PERCENTAGE OF THE WALL IS COVERED BY THE WINDOW? THIS WAS WELL ANSWERED IN THE MAIN. 2.7 / 13.5 × 100 = 20%. QUESTION 2: 10 MARKS (YEARS 10 AND BELOW ONLY) A STORE SELLS REFRIGERATORS WITH A RECOMMENDED PRICE OF $1600. HOWEVER THEY KNOW THAT THEY CANNOT SELL ANY AT THIS PRICE SO THEY OffER THEM AT A ‘SPECIAL’ PRICE OF $1200. (A) WHAT IS THE PERCENTAGE DISCOUNT OF $1200 COMPARED TO $1600? 1200 / 1600 = 75% SO THE ANSWER IS A 25% DISCOUNT. REASONABLE, ALTHOUGH A FEW STUDENTS DIVIDED 1600 BY 1200 AND SOME LEFT 75% AS THEIR ANSWER. (B) IN A BOXING DaY SALE THEY OffER THE FRIDGE AT A SPECIAL ‘20% SAVING’ I.e. 20% Off THE RECOMMENDED PRICE OF $1600. (I) WHAT IS THE SPECIAL BOXING DaY ‘SALE’ PRICe? $1600 - $320 = $1280. WELL ANSWERED IN THE MAIN. THE MOST COMMON MISTAKE WAS TO LEAVE $320 AS THE ଏNAL ANSWER. (II) Does THIS SALE PRICE REPRESENT A SAVING ON THE ‘SPECIAL’ PRICE OR AN INCREAse? YOU DON’T HAVE TO SHOW WORKING. IT’S AN INCREASE SINCE $1280 IS MORE THAN $1200. GENERALLY WELL ANSWERED ALTHOUGH STUDENTS WHO WROTE ‘YES’ OR ‘NO’ WEREN’T GIVING AN ANSWER AT ALL. (C) THEY OffER A ‘FURTHER 20%’ Off THE SALE PRICE IF A CUSTOMER PAYS CASH. WHAT PERCENTAGE SAVING DO THESE TWO PERCENTAGES COMBINED REPRESENt? (1 - 0.8 × 0.8) = 36%. THIS WAS THE ଏRST ‘DIଏCULT’ QUESTION. IF WE TAKE $100 AS THE NORMAL PRICE THEN A 20% DISCOUNT LEADS TO AN $80 SALE PRICE. 20% OF $80 IS $16 SO THE ଏNAL PRICE IS $64, WHICH REPRESENTS A 36% DISCOUNT. SOME STUDENTS WORKED IT OUT. GENERALLY THIS QUESTION WAS A GOOD INDICATOR AS TO WHETHER A STUDENT EVENTUALLY GAINED A MERIT (OR ABOVE) OR NOT. (D) ANOTHER STORE USUALLY MAKES 400% PROଏT ON ITEMS THEY SELL. FOR EXAMPLE AN ITEM COSTING THE STORE $20 WOULD SELL FOR $100. IN A SALE THEY OffER THE $100 ITEM FOR 50% Off THE USUAL PRICe. WHAT PERCENTAGE PROଏT ON THE ITEM DO THEY STILL MAKe? 50% OF $100 IS $50; THUS A $30 PROଏT STILL EXISTS. IF THE ORIGINAL ITEM COST THE STORE $20 THEN THEY STILL MAKE A PROଏT OF 30 / 20 = 150%. MANY STUDENTS WORKED THIS OUT, BUT IT WAS ALSO OFTEN WRONG. 250% AND 30% (OR $30) WERE NOT UNCOMMON. NON-SENSIBLE ANSWERS BIGGER THAN THE ORIGINAL 400% OCCASIONALLY APPEARED. QUESTION 3: 20 MARKS (ALL YEARS) THIS QUESTION WAS FAIRLY ‘EASY’ FOR MANY STUDENTS. FULL CREDIT WAS COMMON. (A) HARRY AND MEGHAN PLAN TO BAKE MUଏNS FOR AFTERNOON TEA. HOWEVER, THE ORIGINAL RECIPE SERVES ଏVE PEOPLe. THEY WILL HAVE TO ADJUST THE AMOUNT OF INGREDIENTS THEY NEED TO SERVE THE TWO OF THEM. (I) THE ORIGINAL RECIPE CALLS FOR 100 G OF BUTTER. HOW MUCH BUTTER IS NEEDED FOR TWO PEOPLe? 2/5 OF 100 IS 40 G. RATIO WAS WELL UNDERSTOOD. (II) THE ORIGINAL RECIPE CALLS FOR 15/16 OF A CUP OF BLUEBERRIES. WHAT FRACTION OF A CUP DO THEY NEED FOR TWO 15/16 PEOPLe? BRIEଏY SHOW YOUR WORKING. 5/2 = 3/8. THERE WERE SEVERAL CANDIDATES WHO REACHED 6/6.4 HERE. THEY SHOWED NO UNDERSTANDING OF DEALING WITH FRACTIONS. (III) HARRY AND MEGHAN WORK OUT THAT THEY NOW NEED TWO CUPS OF ଏOUR. HOW MANY CUPS OF ଏOUR DID THE ORIGINAL RECIPE CALL FOR? 5 CUPS, SINCE 2/5 OF 5 IS 2. (IV) THEY WORK OUT THAT FOR TWO PEOPLE THEY NEED HALF A CUP OF PLAIN YOGHURt. HOW MUCH PLAIN YOGHURT DID THE ORIGINAL RECIPE CALL FOR (I.e. FOR ଏVE PEOPLe)? 0.5 × 5/2 = 1.25 CUPS. A FEW STUDENTS GOT 5/4 BUT LOST CREDIT WHEN THEY GAVE THE ଏNAL ANSWER AS 1.5 CUPS. 2.5 WAS ALSO A ‘COMMON’ MISTAKE. (B) A GOOD RULE OF THUMB FOR CONVERTING OUNCES (OZ) TO GRAMS (G) (ALTHOUGH IT IS NOT PERFECT MATHEMATICALLY) IS TO USE THE FORMULA 1 OZ = 30 G. UsE THIS TO ANSWER THE FOLLOWING TRUE OR FALSE QUESTIONS (YOU DON’T NEED TO SHOW WORKING): (I) 2 G = 60 OZ. FALSE. THIS PROVED TO BE THE HARDEST PART OF THE QUESTION! SEVERAL STUDENTS DIDN’T NOTICE THAT IT WAS ‘BACK-TO-FRONT’. (II) 1.6 OZ = 48 G. TRUE. WELL ANSWERED. (III) 1 KG = 33 OZ (TO THE NEAREST WHOLE NUMBER). TRUE. (1000 / 30 = 33 TO THE NEAREST WHOLE NUMBER.) UNFOR- TUNATELY THIS QUESTION IS AMBIGUOUS AND ‘FALSE’ COULD BE GIVEN BY STUDENTS WHO ’STARTED AT THE RIGHT’. (33 30 = 990, NOT 1000.) THE TROUBLE WITH AMBIGUOUS QUESTIONS IS THAT A FEW STUDENTS ‘WASTE TIME’ ଏGURING× WHICH IS THE ANSWER WE LOOKED FOR. A FEW STUDENTS GAVE BOTH ANSWERS, WITH EXPLANATIONS. THE ANSWER SCHEDULE WAS ADJUSTED FOR THE ERROR. (C) THERE ARE 16 OUNCES (OZ) IN A POUND. REMEMBER THAT WE ARE USING THE APPROXIMATION 1 OZ = 30 G. (I) HOW MANY GRAMS ARE THERE IN 1 POUND? 30 × 16 = 480 G. ALMOST UNIVERSALLY ANSWERED CORRECTLY. (II) HOW MANY POUNDS ARE THERE IN A 6 KG OF ଏOUR? WE WERE LOOKING FOR 6000 / 30 = 200 (OUNCES) AND THEN 200 / 16 = 12.5 (POUNDS). BUT MOST STUDENTS GAVE 6000 / 480 = 12.5 WHICH WAS ଏNE. THE PROBLEM WITH USING ONE OF YOUR OWN ANSWERS IN A LATER PART OF THE QUESTION IS THAT IT MAY BE WRONG, AND SO YOUR SECOND ANSWER BECOMES ‘WRONG’ TOO.
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