q.e for grade 7 (answer key) (1)

8/14/2019 Q.E for Grade 7 (Answer Key) (1)

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Qualifying Exam for grade 7

(Answer Key)

Part A Calculation Skills

1. How many fifths are there in4

5?

Solution:4 1 4

5 45 5 5

= =

2. Express 3.45 as a mixed number in its simplest form.

Solution:45

3.45 3 31!! 2!

= =

3. " 1!# l $ontainer needs 12 full pails of water to fill it $ompletely. %hat is the $apa$ity of the pail?

Solution:1! 5

12 &= l

4. ' $oo(s wor(in) * hours ea$h+ are able to prepare 32 meals. How many $oo(s+ wor(in) at the same speed+

are needed to prepare *4! meals in &h?

Solution: ,raw a table

Cooks ours !eals

' * 321 * 32 - ' 5&

1 1 5& - * '

1 & & / ' 42

? & *4!

*4! - 42 2!

2! $oo(s are needed.

5. 0enneth $y$led from his home to s$hool at an aera)e speed of 15 (mh. "fter s$hool+ his father droe him

home at an aera)e speed of &! (mh. His trip ba$( home too(1

2h less than his trip to s$hool. ind the

distan$e between 0enneths home and his s$hool.Solution: Speed:

Home to s$hool : S$hool to home 15 : &! 1 : 4

ime ta(en:Home to s$hool : S$hool to home 4 : 1

3 units 1

2h

1 unit 1 1

32 &

= h

1

&

h / &! (mh 1! (m

he distan$e between his home and s$hool is 1! (m.

&. ania and Harry are $y$lin) from 6oint " to 6oint 7 whi$h is 52 (m away. ania $y$les at a speed of 12(mh and Harry $y$les at a speed at 1& (mh. How far away will ania be from 6oint 7 when Harry

$ompletes the ra$e?

Solution: 8f ania sets off earlier+ she will be:

4!12 *

&! = (m ahead

Eery hour+ Harry will $at$h by:


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1& 9 12 4 (m

* - 4 2h

e will take "# to catc# u$

'. 8n the fi)ure+ not drawn to s$ale+PQSTis a trapeoid and TP SQ .

TSRis a strai)ht line and QRS is a ri)ht#an)led trian)le. ind

UTSand TUR.

Solution: 6; Sote the number

pattern of exposed fa$es and the number of layers. @1 2 3 4A / 4 2 / 4 / 4 '2 $m2

he area painted )reen is '2 $m2.

P

1&

TS

3*

Q

U

R

6attern 1 6attern 2 6attern 3 6attern 4


3/5

1!. Bhloe+ Ceeta and Daria had some $andies. Bhoe had twi$e as many $andies as Ceeta. %hen Ceeta )ae1

3

of her $andies to Bhloe and Daria )ae5

'of hers to Bhloe+ Bhloe had a total of 4&' $andies. Daria had 5*

$andies left.

Solution: Daria )ae @5* - 2A / 5 145 $andies to Bhloe.' units 4&' 9 145 322

1 unit

322 - ' 4&2 units 2 / 4& 2& units & / 4& 2'&

Part % Pro&lem Sol'ing Skills

1. 5!5 / 2! 5!! / " " " " " ". ind the alue of ".

Solution: 5!5 / 2! 5!!" 5"

"@5!! 5A 5!5"

herefore+ " 2!

2. he len)th of re$tan)le is 2! $m. 8ts width is one fifth its len)th. Express the perimeter of its re$tan)le interms of w in the simplest form.

Solution: %idth1

5!5

x=

4x $m

6erimeter 2!x 2!x 4x 4x

4*x$m

3. he fi)ure below is made up of 2 re$tan)les and suare. 8ts dimensions are )ien in $m.

ind the area of the fi)ure in terms of n.Solution: "rea @n / nA @n / 4A @n ' 9 nA / n

n2 4n 'n

n2 11n n@n 11A

4. " fen$e is to be built one meter away from the sides of a suare field. Ea$h side of the suare is 3f m. ind

the total len)th of the fen$e needed.

Solution: otal len)th needed @3f 1 1A / 4

@12f *A m

5. A+ B+ CandDare four points on a number line representin) the numbers a+ b+ cand d respe$tiely. 8t is

)ien that

Alies on the ri)ht ofB.

n ? '

n

4

n

3f m

1 m

1 m


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Clies on the ri)ht ofDand lies on the left ofA.

Ban cbe the smallest number? Explain your answer.Solution: a > c > d >b or a > b > d or a > b > c > d

Sin$e @any two of the answersA

hus+ no

Part C onroutine Pro&lem Sol'ing

1. he leadin) di)it of a 2!13#di)it number is 5. "ny two nei)hbourin) di)its is either a multiple of 13 or amultiple of 2'. ,etermine the sum of all possible different alues of the last di)it of this 2!13#di)it

number.

Solution:%e (now all the two#di)it numbers that are multiple of 13 are 13+ 2&+ 3+ 52+ &5+ '* and 1.

%hile those two#di)it numbers that are multiple of 2' are 2'+ 54 and *1.

hen the possible first two di)its of this 2!13#di)it number are 52 or 54.his implies the se$ond di)it must be 2 be$ause none of the multiple of 13 or 2' be)in with 4 as the

leadin) di)it.

>ow+ the third di)it is either ' or &. %hen it is '+ it must be followed by * and then the three#$y$le 1+ 3and . he 2!13rd di)it will be 3.

%hen it is &+ then the fourth di)it is 5 a)ain. hus we may hae a three#$y$le 5+ 2 and &+ and the 2!13rddi)it will be &.

Suppose let us swit$h the di)it & to di)it ' at the last step+ then the 2!13rd di)it will be$ome '. Supposewe swit$h earlier+ the 2!13rd di)it will be 3.

hus+ the possible sum is 3 & ' 1&.

2. Cien: 15 151 2 4 .N = + + ,etermine a prime fa$tor ofN.

Solution:

Sin$e( ) ( )15 3! 1545

15 15 3! 15

15 15

2 1 2 2 12 11 2 4 2 2 1

2 1 2 1

+ + + = = = +

and

( ) ( )45 3& 2' 1* 2 1 2 1 2 2 2 2 1 = + + + this implies ( ) ( ) 452 1 2 1 .

7ut ( ) 15 3CB 2 1+ 2 1 2 1+ = and we also (now that

& 3

3

2 12 2 1 '3.

2 1

= + + =

hus+ '3 is one of the prime fa$tor ofN.

3. Fet a, b+c+dbe four distin$t real numbers that satisfy @a +cA@a +dA =@b +cA@b +dA =1. ,etermine the

alue of @a +cA@b +cA.

Solution:

rom the )ien euation+ we hae

@ A@ A @ A@ A !a c a d b c b d + + + + =

this implies2 2@ A @ A@ A !+a b a b c d + + =

that is @ A@ A !.a b a b c d + + + =

7ut +a b then !+a b c d + + + = that isG ( ) +a b c d + = +

so we hae2 2@ A @ A@ A !.a b a b c d + + =

Hen$e+2 21 @ A@ A @ A @ A .a c a d a c d a cd a a b a cd cd ab= + + = + + + = + + =

and2 2@ A@ A @ A @ A 1.a c b c c a b c ab c c d c ab ab cd + + = + + + = + + = =


5/5

4. " suare was partitioned into four $onex poly)on sides without any portion oerlappin). "ssume the

number of sides of these four $onex poly)on as a+ b+ c+ drespe$tiely su$h that 3a b c d > > > . %hi$h of

the followin) 4#tuple positie inte)ers @a+ b+ c+ dA are the possible solution sets?

@&+5+4+3A @'+&+5+4A @'+&+4+3A @'+5+4+3A

".G 7.G B.G ,.G

Solution:

%e $an s(et$h seeral dia)ram and obsere the possible out$omes: we (now that for eery two $onexpoly)on+ there $an hae one $ommon side only+ then there will be a most three sides of eery $onex

poly)on in one suare+ but there will hae at most four sides of $onex poly)on $oin$ide with the side of the)ien suare. Hen$e+ the sides of these $onex poly)on $an not ex$eed 3 4 '. 8t follows that the least

number of side is 3. Hen$e+ the possible 4#tuple positie inte)ers @a+ b+ c+ dA must be either '+ &+ 5+ 4+ or 3.

%hen a '+ then some of the sides in '#sided poly)on will oerlap with two sides of the )ien suare.

Hen$e+ the number of sides of the $onex poly)on must be 2 3 5. + =

hus+ the number of sides of the $onex poly)on $an not be ' and 5. his implies that

the possible 4#tuple positie inte)ers @a+ b+ c+ dA $an only be ( )'+ 5+ 4+ 3 and

( )&+ 5+ 4+ 3 . ;efer the two dia)ram below for a better explanation.

6repared by:

Dr. Simon L. Chua*+E: "ll publi$ations are prote$ted by $opyri)ht. 8t is unlawful to ma(e $opies without the

prior written permission of the Dathemati$s rainers Cuild+ 6hilippines.

q.e for grade 7 (answer key) (1)

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